{"id":"n-816bdb5b69d24e1f","revision":0,"first_index":1,"fingerprint":"f6-11aec07ce60b6d61","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["4","2"],["32","32","8"],["320","480","240","40"],["3584","7168","5376","1792","224"],["43008","107520","107520","53760","13440","1344"],["540672","1622016","2027520","1351680","506880","101376","8448"],["7028736","24600576","36900864","30750720","15375360","4612608","768768","54912"],["93716480","374865920","656015360","656015360","410009600","164003840","41000960","5857280","366080"],["1274544128","5735448576","11470897152","13382713344","10037035008","5018517504","1672839168","358465536","44808192","2489344"],["17611882496","88059412480","198133678080","264178237440","231155957760","138693574656","57788989440","16511139840","3095838720","343982080","17199104"],["246566354944","1356114952192","3390287380480","5085431070720","5085431070720","3559801749504","1779900874752","635678883840","158919720960","26486620160","2648662016","120393728"],["3489862254592","20939173527552","57582727200768","95971212001280","107967613501440","86374090801152","50384886300672","21593522700288","6747975843840","1499550187520","224932528128","20448411648","852017152"],["49855175065600","324058637926400","972175913779200","1782322508595200","2227903135744000","2005112822169600","1336741881446400","668370940723200","250639102771200","69621972992000","13924394598400","1898781081600","158231756800","6085836800"],["717914520944640","5025401646612480","16332555351490560","32665110702981120","44914527216599040","44914527216599040","33685895412449280","19249083092828160","8421473853112320","2807157951037440","701789487759360","127598088683520","15949761085440","1226904698880","43818024960"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","72","1080","18144","326592","6158592","120092544","2401850880","48997757952","1015589892096","21327387734016","452796847276032","9702789584486400","209580255024906240"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-24","-12"],["16","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185533,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-34077ec359de479d"]},{"id":185534,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-34077ec359de479d","e-7698ffbe3b32082b","e-b149ea9150b01cdb"]},{"id":185535,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-34077ec359de479d","e-7698ffbe3b32082b","e-b149ea9150b01cdb"]},{"id":185536,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-34077ec359de479d","e-b149ea9150b01cdb"]},{"id":185537,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7698ffbe3b32082b","e-b149ea9150b01cdb"]}],"region_exact":{},"incoming_edges":[{"id":"e-b149ea9150b01cdb","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-816bdb5b69d24e1f","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"64d8f823e4b6b5102f290e15a8066ae9849952d6813a54c3270e1656f84470f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e8f74cc6dbf727f1","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-34077ec359de479d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-816bdb5b69d24e1f","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"7ab5d460aa2a13f056d8a4efcdd67544d2bdceb981806424e8639faf569f4c7a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e8f74cc6dbf727f1","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-7698ffbe3b32082b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-816bdb5b69d24e1f","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"bda90b24e17ad788636920eed7e1e35f06daa992a0178c28576b6bbed0c79ea7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e69b830457b341cf","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0057a6d9aa35057a","kind":"fingerprint_related","node_a_id":"n-219653841046440e","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-219653841046440e","other":{"id":"n-219653841046440e","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","18"],["27","108","108"]],"card_rows":[["3"],["9","18"],["27","108","108"],["81","486","972","648"]],"row_sums_preview":["3","27","243","2187","19683","177147","1594323","14348907"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610"],"edges":["e-284e6c48cfe1aa93"],"text":"A120909 \u2192 basis A013610"},"headline":"A120909 \u2192 basis A013610","chain":null}},{"id":"l-00ecef4704c8d29e","kind":"fingerprint_related","node_a_id":"n-765f9e2b013b3c68","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-765f9e2b013b3c68","other":{"id":"n-765f9e2b013b3c68","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","6"],["3","12","12"]],"card_rows":[["3"],["3","6"],["3","12","12"],["3","18","36","24"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120909"],"derivation":{"root":"n-765f9e2b013b3c68","steps":[],"edges":[],"text":"A120909"},"headline":"A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","chain":null}},{"id":"l-018b8192011bcaf2","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-31e3d7d6a6f467a8","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-31e3d7d6a6f467a8","other":{"id":"n-31e3d7d6a6f467a8","name":"basis_map_A013609 of derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","4"]],"card_rows":[["-1"],["2","4"],["-3","-12","-12"],["4","24","48","32"]],"row_sums_preview":["0","-1","6","-27","108","-405","1458","-5103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","basis A013609"],"edges":["e-7b1079618314be79","e-3206b568d8163c02"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-0201db0a80ff44d6","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-8e35c880e9f9bcdd","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8e35c880e9f9bcdd","other":{"id":"n-8e35c880e9f9bcdd","name":"derivative of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["32","64"]],"card_rows":[["4"],["32","64"],["288","1152","1152"],["3072","18432","36864","24576"]],"row_sums_preview":["0","4","96","2592","82944","3110400","134369280","6584094720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611","D"],"edges":["e-b8ab5069d7db191f","e-7ba2d9ba397911d9"],"text":"A196347 \u2192 basis A013611 \u2192 D"},"headline":"A196347 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-025e109e3d064c61","kind":"fingerprint_related","node_a_id":"n-3973b74c1ebdbd23","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3973b74c1ebdbd23","other":{"id":"n-3973b74c1ebdbd23","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["21","84","84"]],"card_rows":[["1"],["6","12"],["21","84","84"],["56","336","672","448"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","reverse"],"edges":["e-a52727d7b51dd683","e-16c7d62e82340613"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-04547ee719eef4a2","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-aa18ad1625904333","other":{"id":"n-aa18ad1625904333","name":"mobius_x_over_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["9","36","36"]],"card_rows":[["1"],["4","8"],["9","36","36"],["16","96","192","128"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x/(x+1)"],"edges":["e-963b3afafd9f7f70"],"text":"A135065 \u2192 x \u21a6 x/(x+1)"},"headline":"A135065 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-04c861c2b903481f","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-f047ab7ef03b5b1b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f047ab7ef03b5b1b","other":{"id":"n-f047ab7ef03b5b1b","name":"basis_map_A013611 of basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["96","384","384"]],"card_rows":[["1"],["8","16"],["96","384","384"],["1280","7680","15360","10240"]],"row_sums_preview":["1","24","864","34560","1451520","62705664","2759049216","122974765056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013609","basis A013611"],"edges":["e-2e1c22b656cc26bb","e-973cf9d4b69ecf71"],"text":"A356546 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A356546 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-05027653d6e2db01","kind":"fingerprint_related","node_a_id":"n-66b456586fffaa5f","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-66b456586fffaa5f","other":{"id":"n-66b456586fffaa5f","name":"shift_x_plus_one of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4","4","1"]],"card_rows":[["4","4","1"],["24","36","18","3"],["96","192","144","48","6"],["320","800","800","400","100","10"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x+1"],"edges":["e-ccfad80d3bf960c1","e-7ca9df991456824b"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-0604e45f43854a01","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-50bfbc2b3dd01b9f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-50bfbc2b3dd01b9f","other":{"id":"n-50bfbc2b3dd01b9f","name":"basis_map_A013610 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["72","288","288"]],"card_rows":[["1"],["6","12"],["72","288","288"],["1080","6480","12960","8640"]],"row_sums_preview":["1","18","648","29160","1469664","79361856","4489613568","262642393728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611","basis A013610"],"edges":["e-9f3d1c5f0f510549","e-18017ec3549f8c68"],"text":"A085880 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A085880 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-061a6f00164478de","kind":"fingerprint_related","node_a_id":"n-e04be2527aa35149","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e04be2527aa35149","other":{"id":"n-e04be2527aa35149","name":"mobius_x_over_x_plus_one of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","24"]],"card_rows":[["1"],["2","4"],["6","24","24"],["20","120","240","160"]],"row_sums_preview":["1","6","54","540","5670","61236","673596","7505784"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x/(x+1)"],"edges":["e-5e9e426586f7e874"],"text":"A356546 \u2192 x \u21a6 x/(x+1)"},"headline":"A356546 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-06d61c5af8f21d4a","kind":"fingerprint_related","node_a_id":"n-fe7b03c8bdb96043","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fe7b03c8bdb96043","other":{"id":"n-fe7b03c8bdb96043","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","5"],["60","60","15"]],"card_rows":[["1"],["10","5"],["60","60","15"],["280","420","210","35"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318"],"edges":["e-484d449f4d0f99c5"],"text":"A178821 \u2192 basis A007318"},"headline":"A178821 \u2192 basis A007318","chain":null}},{"id":"l-07dc5ba67b215fcb","kind":"fingerprint_related","node_a_id":"n-af521840f49b6d03","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af521840f49b6d03","other":{"id":"n-af521840f49b6d03","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","6"]],"card_rows":[["3"],["12","6"],["36","36","9"],["96","144","72","12"]],"row_sums_preview":["0","3","18","81","324","1215","4374","15309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D"],"edges":["e-d19cf3982d12ec2e"],"text":"A120910 \u2192 D"},"headline":"A120910 \u2192 D","chain":null}},{"id":"l-0811be10c62ce9bf","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-1420b73426c65765","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1420b73426c65765","other":{"id":"n-1420b73426c65765","name":"basis_map_A013611 of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","32"],["160","640","640"]],"card_rows":[["1"],["16","32"],["160","640","640"],["1280","7680","15360","10240"]],"row_sums_preview":["1","48","1440","34560","725760","13934592","250822656","4299816960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013609","basis A013611"],"edges":["e-9fe823de1e8617e8","e-46c076477c7812ae"],"text":"A178820 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A178820 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-08124d5b127ca80c","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-e8f0ed2d58ae5d84","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e8f0ed2d58ae5d84","other":{"id":"n-e8f0ed2d58ae5d84","name":"basis_map_A013610 of derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["12","24"]],"card_rows":[["2"],["12","24"],["54","216","216"],["216","1296","2592","1728"]],"row_sums_preview":["0","2","36","486","5832","65610","708588","7440174"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D","basis A013610"],"edges":["e-9f3f778aa1812acb","e-3fb3eef32d402839"],"text":"A013609 \u2192 D \u2192 basis A013610"},"headline":"A013609 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-094473d4e4973ef5","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-a535589f024a4075","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a535589f024a4075","other":{"id":"n-a535589f024a4075","name":"basis_map_A013611 of shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","16","16"],["8","48","96","64"],["16","128","384","512","256"]],"card_rows":[["4","16","16"],["8","48","96","64"],["16","128","384","512","256"],["32","320","1280","2560","2560","1024"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["x \u21a6 x+1","basis A013611"],"edges":["e-cb9b2e4b7b317e85","e-9b0ff81e6934e8b6"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A013611"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A013611","chain":null}},{"id":"l-0a030a718344ee2f","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-83e01962f87310bb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-83e01962f87310bb","other":{"id":"n-83e01962f87310bb","name":"derivative of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["48","96"]],"card_rows":[["4"],["48","96"],["720","2880","2880"],["13440","80640","161280","107520"]],"row_sums_preview":["0","4","144","6480","362880","24494400","1939956480","176536039680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","D"],"edges":["e-20dcbad557ba463b","e-ef90917cc64c6244"],"text":"A085881 \u2192 basis A013611 \u2192 D"},"headline":"A085881 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-0a2f21b31869310e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-0df7e30176c773f9","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0df7e30176c773f9","other":{"id":"n-0df7e30176c773f9","name":"basis_map_A013609 of Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["4","24","48","32"]],"row_sums_preview":["1","3","18","108","648","3888","23328","139968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a895a528c93c18ea","steps":["basis A013609"],"edges":["e-9e5cdc58bf7cd01f"],"text":"A134309 \u2192 basis A013609"},"headline":"A134309 \u2192 basis A013609","chain":null}},{"id":"l-0b28d09ae11863ce","kind":"fingerprint_related","node_a_id":"n-75aa1f320970e052","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-75aa1f320970e052","other":{"id":"n-75aa1f320970e052","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["10","40","40"]],"card_rows":[["1"],["4","8"],["10","40","40"],["20","120","240","160"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","reverse"],"edges":["e-2fa6de2302a1c958","e-a34a2dedb4e4f379"],"text":"A178820 \u2192 basis A007318 \u2192 reverse"},"headline":"A178820 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-0ba3ac2636158e75","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-660b73dc06a54453","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-660b73dc06a54453","other":{"id":"n-660b73dc06a54453","name":"basis_map_A013609 of theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","12","12"],["4","24","48","32"],["5","40","120","160","80"]],"card_rows":[["3","12","12"],["4","24","48","32"],["5","40","120","160","80"],["6","60","240","480","480","192"]],"row_sums_preview":["27","108","405","1458","5103","17496","59049","196830"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["\u03b8+1","basis A013609"],"edges":["e-972f6eabe8338070","e-a784a1b91a6a53a9"],"text":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-0c2b81632ac57d21","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-75e1d8aa643b4cef","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-75e1d8aa643b4cef","other":{"id":"n-75e1d8aa643b4cef","name":"basis_map_A013610 of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["216","864","864"]],"card_rows":[["1"],["12","24"],["216","864","864"],["5184","31104","62208","41472"]],"row_sums_preview":["1","36","1944","139968","12597120","1360488960","171421608960","24684711690240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611","basis A013610"],"edges":["e-413ab29272bdfecc","e-2003c2a9d5b8a3f9"],"text":"A156992 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-0c6760396f855679","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-7e687fdd41bd2689","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7e687fdd41bd2689","other":{"id":"n-7e687fdd41bd2689","name":"reversal of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","22"],["121","484","484"]],"card_rows":[["1"],["11","22"],["121","484","484"],["1331","7986","15972","10648"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","reverse"],"edges":["e-a5d0659f5315d3b7","e-b378c241b83c7af0"],"text":"A038325 \u2192 basis A007318 \u2192 reverse"},"headline":"A038325 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-0cb7a14a958ee540","kind":"fingerprint_related","node_a_id":"n-0b806d8871f729f4","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0b806d8871f729f4","other":{"id":"n-0b806d8871f729f4","name":"mobius_x_over_x_plus_one of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","12"]],"card_rows":[["1"],["1","2"],["3","12","12"],["7","42","84","56"]],"row_sums_preview":["1","3","27","189","1539","12393","102789","859491"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["x \u21a6 x/(x+1)"],"edges":["e-97db3bd73c62b68d"],"text":"A376827 \u2192 x \u21a6 x/(x+1)"},"headline":"A376827 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0d7944a3000d33ca","kind":"fingerprint_related","node_a_id":"n-4a6b9237eddcc625","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4a6b9237eddcc625","other":{"id":"n-4a6b9237eddcc625","name":"mobius_x_over_x_plus_one of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","12"]],"card_rows":[["1"],["1","2"],["3","12","12"],["5","30","60","40"]],"row_sums_preview":["1","3","27","135","891","5103","31347","185895"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["x \u21a6 x/(x+1)"],"edges":["e-33ed1f696f0d9ad5"],"text":"A124860 \u2192 x \u21a6 x/(x+1)"},"headline":"A124860 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0f3241e6336ea69b","kind":"fingerprint_related","node_a_id":"n-ed2a02208588b66b","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ed2a02208588b66b","other":{"id":"n-ed2a02208588b66b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["21","84","84"]],"card_rows":[["1"],["6","12"],["21","84","84"],["56","336","672","448"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x/(x+1)"],"edges":["e-376fcdbb8dac8b5b"],"text":"A178822 \u2192 x \u21a6 x/(x+1)"},"headline":"A178822 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-119c893285ac7736","kind":"fingerprint_related","node_a_id":"n-11dd292935cd7dbf","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-11dd292935cd7dbf","other":{"id":"n-11dd292935cd7dbf","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["10","40","40"]],"card_rows":[["1"],["4","8"],["10","40","40"],["20","120","240","160"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["x \u21a6 x/(x+1)"],"edges":["e-384d0b9c26a615c7"],"text":"A178820 \u2192 x \u21a6 x/(x+1)"},"headline":"A178820 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-127600e10007accc","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-df90e6e2770b9985","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-df90e6e2770b9985","other":{"id":"n-df90e6e2770b9985","name":"derivative of basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["24","12"]],"card_rows":[["2"],["24","12"],["288","288","72"],["3840","5760","2880","480"]],"row_sums_preview":["0","2","36","648","12960","291600","7348320","205752960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318","D"],"edges":["e-bd56c468ffc76132","e-d59f0db741671966"],"text":"A156992 \u2192 basis A007318 \u2192 D"},"headline":"A156992 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-131b64ff34b3d8b0","kind":"fingerprint_related","node_a_id":"n-df44f407d23e429d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-df44f407d23e429d","other":{"id":"n-df44f407d23e429d","name":"basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["192","288","144","24"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318"],"edges":["e-bd56c468ffc76132"],"text":"A156992 \u2192 basis A007318"},"headline":"A156992 \u2192 basis A007318","chain":null}},{"id":"l-1359ce726ff4b9da","kind":"fingerprint_related","node_a_id":"n-d34847103847793f","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-d34847103847793f","other":{"id":"n-d34847103847793f","name":"shift_x_plus_one of basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-2"],["16","16","4"]],"card_rows":[["1"],["-4","-2"],["16","16","4"],["-64","-96","-48","-8"]],"row_sums_preview":["1","-6","36","-216","1296","-7776","46656","-279936"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-b6cdc8ab0f2a09ef","e-ae40ead3d3d3dc2f"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-13b2c6ea3e9910a6","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-428355ff004632e5","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-428355ff004632e5","other":{"id":"n-428355ff004632e5","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["64","128"]],"card_rows":[["8"],["64","128"],["384","1536","1536"],["2048","12288","24576","16384"]],"row_sums_preview":["0","8","192","3456","55296","829440","11943936","167215104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611","D"],"edges":["e-e6fcabb7b5383fc4","e-756289e629e2ab28"],"text":"A038208 \u2192 basis A013611 \u2192 D"},"headline":"A038208 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-13c51bb768a73802","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7c4e320f16a7ccb1","other":{"id":"n-7c4e320f16a7ccb1","name":"shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["12","12","3"],["32","48","24","4"]],"row_sums_preview":["0","1","6","27","108","405","1458","5103"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["x \u21a6 x+1"],"edges":["e-1a916fd0c60fb95f"],"text":"A003506 \u2192 x \u21a6 x+1"},"headline":"A003506 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-14625d18882b3e2e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-53cc0bcdc05addd6","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-53cc0bcdc05addd6","other":{"id":"n-53cc0bcdc05addd6","name":"basis_map_A013609 of derivative of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","4"]],"card_rows":[["1"],["8","4"],["48","48","12"],["256","384","192","32"]],"row_sums_preview":["0","1","12","108","864","6480","46656","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["D","basis A013609"],"edges":["e-6602f7f73d905d9e","e-38ae1e9cdcd07e7a"],"text":"A027465 \u2192 D \u2192 basis A013609"},"headline":"A027465 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-14cb971c4db14480","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-285ca2c8fd4bd5fc","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-285ca2c8fd4bd5fc","other":{"id":"n-285ca2c8fd4bd5fc","name":"basis_map_A013609 of theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["3","12","12"]],"card_rows":[["1"],["-2","-4"],["3","12","12"],["-4","-24","-48","-32"]],"row_sums_preview":["1","-6","27","-108","405","-1458","5103","-17496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["\u03b8+1","basis A013609"],"edges":["e-5a10774a37bfecce","e-c932186c012e452c"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-167114500ce86e1f","kind":"fingerprint_related","node_a_id":"n-2db2a1c44af41062","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2db2a1c44af41062","other":{"id":"n-2db2a1c44af41062","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"]],"card_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"],["7962624","19906560","19906560","9953280","2488320","248832"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1"],"edges":["e-240272ddc3b4d83d"],"text":"A038338 \u2192 x \u21a6 x+1"},"headline":"A038338 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-1899a71c389d3115","kind":"fingerprint_related","node_a_id":"n-fde1ec7f640113ad","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fde1ec7f640113ad","other":{"id":"n-fde1ec7f640113ad","name":"derivative of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["24","12"]],"card_rows":[["3"],["24","12"],["120","120","30"],["480","720","360","60"]],"row_sums_preview":["0","3","36","270","1620","8505","40824","183708"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","D"],"edges":["e-34e3d9a08d46fb54","e-e3792d909ca1de06"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-1a723b5104828692","kind":"fingerprint_related","node_a_id":"n-890099793b4c816b","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-890099793b4c816b","other":{"id":"n-890099793b4c816b","name":"mobius_x_over_x_plus_one of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"]],"card_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"],["0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x/(x+1)"],"edges":["e-dd3ac95d772228bd"],"text":"A159854 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1ad3067385ae4db1","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5e0aee9b1616fa35","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5e0aee9b1616fa35","other":{"id":"n-5e0aee9b1616fa35","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["216","864","864"]],"card_rows":[["1"],["12","24"],["216","864","864"],["4320","25920","51840","34560"]],"row_sums_preview":["1","36","1944","116640","7348320","476171136","31427294976","2101139149824"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","basis A013610"],"edges":["e-6d76bdb8087cd7e0","e-4811d36cbd9b8caa"],"text":"A356546 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A356546 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-1afd0f8cf21dd9fe","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-42eb612371a67948","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-42eb612371a67948","other":{"id":"n-42eb612371a67948","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2304","2304","576"],["110592","165888","82944","13824"],["5308416","10616832","7962624","2654208","331776"]],"card_rows":[["2304","2304","576"],["110592","165888","82944","13824"],["5308416","10616832","7962624","2654208","331776"],["254803968","637009920","637009920","318504960","79626240","7962624"]],"row_sums_preview":["5184","373248","26873856","1934917632","139314069504","10030613004288","722204136308736","51998697814228992"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609","basis A007318"],"edges":["e-9ec880266e74c89d","e-5470a30744a2e5b2"],"text":"A038338 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038338 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-1b4a57692a219c3e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-63972a365a111569","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-63972a365a111569","other":{"id":"n-63972a365a111569","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","576","576"],["1728","10368","20736","13824"],["20736","165888","497664","663552","331776"]],"card_rows":[["144","576","576"],["1728","10368","20736","13824"],["20736","165888","497664","663552","331776"],["248832","2488320","9953280","19906560","19906560","7962624"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x/(x+1)"],"edges":["e-3e7423bb54bbe99c"],"text":"A038338 \u2192 x \u21a6 x/(x+1)"},"headline":"A038338 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1bd6370b82693ff8","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-f70d1cd881945afe","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f70d1cd881945afe","other":{"id":"n-f70d1cd881945afe","name":"basis_map_A013611 of Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["8","32","32"]],"card_rows":[["1"],["4","8"],["8","32","32"],["16","96","192","128"]],"row_sums_preview":["1","12","72","432","2592","15552","93312","559872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f70486c726c7ad1c","steps":["basis A013611"],"edges":["e-0b9fee1e1f489294"],"text":"A134058 \u2192 basis A013611"},"headline":"A134058 \u2192 basis A013611","chain":null}},{"id":"l-1d189df6c7ef8ee5","kind":"fingerprint_related","node_a_id":"n-bc7333ad044cbf73","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bc7333ad044cbf73","other":{"id":"n-bc7333ad044cbf73","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["64","64","16"]],"card_rows":[["1"],["8","4"],["64","64","16"],["512","768","384","64"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611"],"edges":["e-267942f18014ef6e"],"text":"A027466 \u2192 basis A013611"},"headline":"A027466 \u2192 basis A013611","chain":null}},{"id":"l-1d8285406c098400","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-abd3b95d2350f264","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-abd3b95d2350f264","other":{"id":"n-abd3b95d2350f264","name":"basis_map_A013609 of derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["18","36"]],"card_rows":[["4"],["18","36"],["48","192","192"],["100","600","1200","800"]],"row_sums_preview":["0","4","54","432","2700","14580","71442","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D","basis A013609"],"edges":["e-06738ea5bb413735","e-864c354a3903aafb","e-63516dcc6ed9426a"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 basis A013609"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-1ea936fb94ee9382","kind":"fingerprint_related","node_a_id":"n-8d4637810467d09a","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8d4637810467d09a","other":{"id":"n-8d4637810467d09a","name":"basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["40","60","30","5"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1"],"edges":["e-45645d17dad7cf5c"],"text":"A085880 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-2150eebac8c89689","kind":"fingerprint_related","node_a_id":"n-feee915c2922c082","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-feee915c2922c082","other":{"id":"n-feee915c2922c082","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["36","144","144"]],"card_rows":[["1"],["6","12"],["36","144","144"],["216","1296","2592","1728"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611"],"edges":["e-fc7ebcb9132c171a"],"text":"A038221 \u2192 basis A013611"},"headline":"A038221 \u2192 basis A013611","chain":null}},{"id":"l-23356087d1a3cc7c","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ec6eba6199dd71c6","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec6eba6199dd71c6","other":{"id":"n-ec6eba6199dd71c6","name":"basis_map_A013611 of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A007318","basis A013611"],"edges":["e-279353e75dd63f59","e-8cd116134dcaa22c"],"text":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 basis A013611","chain":null}},{"id":"l-2532e07c748148fc","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-3ef6ce68fe5dae6d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3ef6ce68fe5dae6d","other":{"id":"n-3ef6ce68fe5dae6d","name":"basis_map_A013611 of Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["12","48","48"]],"card_rows":[["1"],["6","12"],["12","48","48"],["24","144","288","192"]],"row_sums_preview":["1","18","108","648","3888","23328","139968","839808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f71648c3ccd02d4","steps":["basis A013611"],"edges":["e-f8de2ea15957ae49"],"text":"A134059 \u2192 basis A013611"},"headline":"A134059 \u2192 basis A013611","chain":null}},{"id":"l-25afd498b22b1374","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-c31555cef32b0bd7","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c31555cef32b0bd7","other":{"id":"n-c31555cef32b0bd7","name":"basis_map_A013611 of derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["12","24"],["48","192","192"],["160","960","1920","1280"]],"row_sums_preview":["0","0","2","36","432","4320","38880","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["D","basis A013611"],"edges":["e-0d0bdaee7e5f71e7","e-e9bef020609311e9"],"text":"A003506 \u2192 D \u2192 basis A013611"},"headline":"A003506 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-270cf525f1d39e10","kind":"fingerprint_related","node_a_id":"n-5aec7bf3521b85d9","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5aec7bf3521b85d9","other":{"id":"n-5aec7bf3521b85d9","name":"derivative of derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["6"]],"card_rows":[["6"],["36","18"],["144","144","36"],["480","720","360","60"]],"row_sums_preview":["0","0","6","54","324","1620","7290","30618"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D","D"],"edges":["e-d19cf3982d12ec2e","e-d0b8f89db9f3bf8e"],"text":"A120910 \u2192 D \u2192 D"},"headline":"A120910 \u2192 D \u2192 D","chain":null}},{"id":"l-2790b1c8d15b0818","kind":"fingerprint_related","node_a_id":"n-77621b0cfa8249dd","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-77621b0cfa8249dd","other":{"id":"n-77621b0cfa8249dd","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["40","160","160"]],"card_rows":[["1"],["8","16"],["40","160","160"],["160","960","1920","1280"]],"row_sums_preview":["1","24","360","4320","45360","435456","3919104","33592320"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611"],"edges":["e-159425c642d412bb"],"text":"A178820 \u2192 basis A013611"},"headline":"A178820 \u2192 basis A013611","chain":null}},{"id":"l-287e32d37fe7747c","kind":"fingerprint_related","node_a_id":"n-1aea442b4a268aa3","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1aea442b4a268aa3","other":{"id":"n-1aea442b4a268aa3","name":"derivative of derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["8"]],"card_rows":[["8"],["24","48"],["48","192","192"],["80","480","960","640"]],"row_sums_preview":["0","0","8","72","432","2160","9720","40824"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D","D"],"edges":["e-9f3f778aa1812acb","e-17fc9e98414a9fae"],"text":"A013609 \u2192 D \u2192 D"},"headline":"A013609 \u2192 D \u2192 D","chain":null}},{"id":"l-28b4cd46ce2e310b","kind":"fingerprint_related","node_a_id":"n-f6cae63539e48dd7","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f6cae63539e48dd7","other":{"id":"n-f6cae63539e48dd7","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","12"],["336","336","84"]],"card_rows":[["1"],["24","12"],["336","336","84"],["3584","5376","2688","448"]],"row_sums_preview":["1","36","756","12096","163296","1959552","21555072","221709312"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-5270021ea4ce6d61","e-3c043d586e042617"],"text":"A178822 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-291a6c6e4f0727f0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-6014e4c463d6a6b1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6014e4c463d6a6b1","other":{"id":"n-6014e4c463d6a6b1","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["48","48","12"]],"card_rows":[["1"],["4","2"],["48","48","12"],["960","1440","720","120"]],"row_sums_preview":["1","6","108","3240","136080","7348320","484989120","37829151360"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","reverse"],"edges":["e-20dcbad557ba463b","e-07ec6c4313f230fd"],"text":"A085881 \u2192 basis A013611 \u2192 reverse"},"headline":"A085881 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-2edc27d6b5b86d1a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-b10dbc4af39a08bb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b10dbc4af39a08bb","other":{"id":"n-b10dbc4af39a08bb","name":"derivative of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["11"],["484","242"]],"card_rows":[["11"],["484","242"],["15972","15972","3993"],["468512","702768","351384","58564"]],"row_sums_preview":["0","11","726","35937","1581228","65225655","2582935938","99443033613"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","D"],"edges":["e-a5d0659f5315d3b7","e-a6fc9f345a420dd6"],"text":"A038325 \u2192 basis A007318 \u2192 D"},"headline":"A038325 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-2f957d6e044a1983","kind":"fingerprint_related","node_a_id":"n-fdb3be84f583ed95","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fdb3be84f583ed95","other":{"id":"n-fdb3be84f583ed95","name":"basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"]],"card_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"],["1","10","40","80","80","32"]],"row_sums_preview":["9","27","81","243","729","2187","6561","19683"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609"],"edges":["e-7bee75fc05394bb8"],"text":"basis-map coefficients of A112227 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609","chain":null}},{"id":"l-323ba8aaeaba1846","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2447747166f6803a","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2447747166f6803a","other":{"id":"n-2447747166f6803a","name":"basis_map_A013609 of derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["14","28"]],"card_rows":[["4"],["14","28"],["30","120","120"],["52","312","624","416"]],"row_sums_preview":["0","4","42","270","1404","6480","27702","112266"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D","basis A013609"],"edges":["e-7e9b074076a3cdbe","e-684bd95aaeb80906"],"text":"A131431 \u2192 D \u2192 basis A013609"},"headline":"A131431 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-323e0a40059dd5c6","kind":"fingerprint_related","node_a_id":"n-0ea1c8d30da3abae","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0ea1c8d30da3abae","other":{"id":"n-0ea1c8d30da3abae","name":"mobius_x_over_x_plus_one of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1","4","4"]],"card_rows":[["1","4","4"],["3","18","36","24"],["6","48","144","192","96"],["10","100","400","800","800","320"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-ccfad80d3bf960c1","e-018032959df2ae1c"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3330f3fa64d66f21","kind":"fingerprint_related","node_a_id":"n-cc4620131f7b034d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-cc4620131f7b034d","other":{"id":"n-cc4620131f7b034d","name":"shift_x_plus_one of derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["12","6"],["48","48","12"],["160","240","120","20"]],"row_sums_preview":["0","0","2","18","108","540","2430","10206"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["D","x \u21a6 x+1"],"edges":["e-0d0bdaee7e5f71e7","e-5a7dc1155b797dbc"],"text":"A003506 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A003506 \u2192 D \u2192 x \u21a6 x+1","chain":null}},{"id":"l-34b4fa9900fb21fe","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-abded0ce0d665ea5","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-abded0ce0d665ea5","other":{"id":"n-abded0ce0d665ea5","name":"basis_map_A013610 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["108","432","432"]],"card_rows":[["1"],["6","12"],["108","432","432"],["3240","19440","38880","25920"]],"row_sums_preview":["1","18","972","87480","11022480","1785641760","353557068480","82732354024320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","basis A013610"],"edges":["e-20dcbad557ba463b","e-fc36264f056d4b8e"],"text":"A085881 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A085881 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-36db1eb77ee7797e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5245c7f95f442806","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5245c7f95f442806","other":{"id":"n-5245c7f95f442806","name":"basis_map_A013611 of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["28","112","112"]],"card_rows":[["1"],["8","16"],["28","112","112"],["80","480","960","640"]],"row_sums_preview":["1","24","252","2160","16848","124416","886464","6158592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","basis A013611"],"edges":["e-28824f426538156a","e-f93f7e05bbc00847"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A013611"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A013611","chain":null}},{"id":"l-3994d3889206001e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-0ee74d387e65778a","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0ee74d387e65778a","other":{"id":"n-0ee74d387e65778a","name":"basis_map_A013609 of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["7","28","28"]],"card_rows":[["1"],["3","6"],["7","28","28"],["15","90","180","120"]],"row_sums_preview":["1","9","63","405","2511","15309","92583","557685"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["basis A013609"],"edges":["e-92441702a47c4c4d"],"text":"A131436 \u2192 basis A013609"},"headline":"A131436 \u2192 basis A013609","chain":null}},{"id":"l-3a9c5d5c0938d8a8","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-8a3a266f7cc7ab95","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8a3a266f7cc7ab95","other":{"id":"n-8a3a266f7cc7ab95","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","1024","1024"],["4096","24576","49152","32768"],["65536","524288","1572864","2097152","1048576"]],"card_rows":[["256","1024","1024"],["4096","24576","49152","32768"],["65536","524288","1572864","2097152","1048576"],["1048576","10485760","41943040","83886080","83886080","33554432"]],"row_sums_preview":["2304","110592","5308416","254803968","12230590464","587068342272","28179280429056","1352605460594688"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-dab838a9194ad2c3","e-38fdfef9bf874526"],"text":"A038330 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038330 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3aff91dde6207ea8","kind":"fingerprint_related","node_a_id":"n-57f251ac09effa9f","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-57f251ac09effa9f","other":{"id":"n-57f251ac09effa9f","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","20"],["60","240","240"]],"card_rows":[["1"],["10","20"],["60","240","240"],["280","1680","3360","2240"]],"row_sums_preview":["1","30","540","7560","90720","979776","9797760","92378880"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611"],"edges":["e-4faaf8fb320b6fca"],"text":"A178821 \u2192 basis A013611"},"headline":"A178821 \u2192 basis A013611","chain":null}},{"id":"l-3bbefe161fc8756b","kind":"fingerprint_related","node_a_id":"n-fbc103865e18ab22","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fbc103865e18ab22","other":{"id":"n-fbc103865e18ab22","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","10"],["15","60","60"]],"card_rows":[["1"],["5","10"],["15","60","60"],["35","210","420","280"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["x \u21a6 x/(x+1)"],"edges":["e-c12420fb71b76e58"],"text":"A178821 \u2192 x \u21a6 x/(x+1)"},"headline":"A178821 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3c2b0af7e794301a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-3b43f24ee4a65071","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3b43f24ee4a65071","other":{"id":"n-3b43f24ee4a65071","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["64","32"]],"card_rows":[["4"],["64","32"],["768","768","192"],["8192","12288","6144","1024"]],"row_sums_preview":["0","4","96","1728","27648","414720","5971968","83607552"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611","D"],"edges":["e-267942f18014ef6e","e-d04f2849f39db31c"],"text":"A027466 \u2192 basis A013611 \u2192 D"},"headline":"A027466 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-3cebf6e62e9586a3","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-8a855024e6ba6660","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8a855024e6ba6660","other":{"id":"n-8a855024e6ba6660","name":"basis_map_A013609 of derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","12"],["12","48","48"],["20","120","240","160"]],"row_sums_preview":["0","0","2","18","108","540","2430","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","basis A013609"],"edges":["e-1aec373ef809ab0b","e-ec86f423e6dc286e"],"text":"A127648 \u2192 D \u2192 basis A013609"},"headline":"A127648 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-3f117bb2e7199ea3","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-165dae7ec5d72aaf","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-165dae7ec5d72aaf","other":{"id":"n-165dae7ec5d72aaf","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","1024","256"],["32768","49152","24576","4096"],["1048576","2097152","1572864","524288","65536"]],"card_rows":[["1024","1024","256"],["32768","49152","24576","4096"],["1048576","2097152","1572864","524288","65536"],["33554432","83886080","83886080","41943040","10485760","1048576"]],"row_sums_preview":["2304","110592","5308416","254803968","12230590464","587068342272","28179280429056","1352605460594688"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611","basis A007318"],"edges":["e-dab838a9194ad2c3","e-0f1f659f8a53aaaa"],"text":"A038330 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-40052f1975050a49","kind":"fingerprint_related","node_a_id":"n-c632c15f8444f2a5","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c632c15f8444f2a5","other":{"id":"n-c632c15f8444f2a5","name":"derivative of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","8"]],"card_rows":[["2"],["4","8"],["6","24","24"],["8","48","96","64"]],"row_sums_preview":["0","2","12","54","216","810","2916","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","D"],"edges":["e-7869781294d5f3a8","e-5badf073aff000c5"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-4031a87286db1a82","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-b13c43ab7c0b4b74","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b13c43ab7c0b4b74","other":{"id":"n-b13c43ab7c0b4b74","name":"derivative of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","16"]],"card_rows":[["2"],["8","16"],["36","144","144"],["192","1152","2304","1536"]],"row_sums_preview":["0","2","24","324","5184","97200","2099520","51438240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","D"],"edges":["e-456dc355a6d78eff","e-4294cfd5037177a3"],"text":"A136572 \u2192 basis A013609 \u2192 D"},"headline":"A136572 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-41b0039d34294711","kind":"fingerprint_related","node_a_id":"n-db0f883a735747d6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-db0f883a735747d6","other":{"id":"n-db0f883a735747d6","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["18","9"],["108","108","27"]],"card_rows":[["3"],["18","9"],["108","108","27"],["648","972","486","81"]],"row_sums_preview":["3","27","243","2187","19683","177147","1594323","14348907"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","reverse"],"edges":["e-284e6c48cfe1aa93","e-99b66bb3dfeecb18"],"text":"A120909 \u2192 basis A013610 \u2192 reverse"},"headline":"A120909 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-441a68cb9cea79df","kind":"fingerprint_related","node_a_id":"n-986a623282b41276","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-986a623282b41276","other":{"id":"n-986a623282b41276","name":"basis_map_A013609 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["2","8","8"]],"card_rows":[["1"],["-1","-2"],["2","8","8"],["-6","-36","-72","-48"]],"row_sums_preview":["1","-3","18","-162","1944","-29160","524880","-11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013609"],"edges":["e-848f5dbb927a8d76"],"text":"basis-map coefficients of A021012 \u2192 basis A013609"},"headline":"basis-map coefficients of A021012 \u2192 basis A013609","chain":null}},{"id":"l-456986e2abbd7c1d","kind":"fingerprint_related","node_a_id":"n-af416c91e2aef37c","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af416c91e2aef37c","other":{"id":"n-af416c91e2aef37c","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["24","24","6"]],"card_rows":[["1"],["6","3"],["24","24","6"],["80","120","60","10"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1"],"edges":["e-34e3d9a08d46fb54"],"text":"A094305 \u2192 x \u21a6 x+1"},"headline":"A094305 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-4851510b47471e1a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ec2d40ae4d2a6c1e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec2d40ae4d2a6c1e","other":{"id":"n-ec2d40ae4d2a6c1e","name":"reversal of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","12"]],"card_rows":[["1"],["1","2"],["3","12","12"],["15","90","180","120"]],"row_sums_preview":["1","3","27","405","8505","229635","7577955","295540245"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","reverse"],"edges":["e-5048ce3ebd740a6f","e-978716e8aa870847"],"text":"A085881 \u2192 basis A007318 \u2192 reverse"},"headline":"A085881 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-489a5a12889b17f1","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-66bf587f689cc703","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-66bf587f689cc703","other":{"id":"n-66bf587f689cc703","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["12"],["144","288"]],"card_rows":[["12"],["144","288"],["1296","5184","5184"],["10368","62208","124416","82944"]],"row_sums_preview":["0","12","432","11664","279936","6298560","136048896","2857026816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","D"],"edges":["e-fc7ebcb9132c171a","e-4b112dcc96d6b521"],"text":"A038221 \u2192 basis A013611 \u2192 D"},"headline":"A038221 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-49907c56b67f2643","kind":"fingerprint_related","node_a_id":"n-ec536fa823a7f7aa","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec536fa823a7f7aa","other":{"id":"n-ec536fa823a7f7aa","name":"derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","8"]],"card_rows":[["2"],["4","8"],["6","24","24"],["8","48","96","64"]],"row_sums_preview":["0","2","12","54","216","810","2916","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D"],"edges":["e-9f3f778aa1812acb"],"text":"A013609 \u2192 D"},"headline":"A013609 \u2192 D","chain":null}},{"id":"l-4bff6da139339448","kind":"fingerprint_related","node_a_id":"n-1e156f12069736a3","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1e156f12069736a3","other":{"id":"n-1e156f12069736a3","name":"basis_map_A013611 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["4","16","16"]],"card_rows":[["1"],["-2","-4"],["4","16","16"],["-8","-48","-96","-64"]],"row_sums_preview":["1","-6","36","-216","1296","-7776","46656","-279936"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013611"],"edges":["e-6ef42593efb6357c"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013611","chain":null}},{"id":"l-4d32637bbd0960c2","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-593ccaf7119eeddb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-593ccaf7119eeddb","other":{"id":"n-593ccaf7119eeddb","name":"basis_map_A013610 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-6"],["9","36","36"]],"card_rows":[["1"],["-3","-6"],["9","36","36"],["-27","-162","-324","-216"]],"row_sums_preview":["1","-9","81","-729","6561","-59049","531441","-4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","basis A013610"],"edges":["e-66f8525b021adc45","e-42915567a929bf14"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-4e0af5aace96f985","kind":"fingerprint_related","node_a_id":"n-869baea9cb22bcd6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-869baea9cb22bcd6","other":{"id":"n-869baea9cb22bcd6","name":"derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["84","42"]],"card_rows":[["6"],["84","42"],["672","672","168"],["4032","6048","3024","504"]],"row_sums_preview":["0","6","126","1512","13608","102060","673596","4041576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","D"],"edges":["e-a52727d7b51dd683","e-c40bf927f673dbd9"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-4e47b1d593c1b727","kind":"fingerprint_related","node_a_id":"n-53ad300b8c449980","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-53ad300b8c449980","other":{"id":"n-53ad300b8c449980","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["9","36","36"]],"card_rows":[["1"],["3","6"],["9","36","36"],["27","162","324","216"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610"],"edges":["e-14a3a792ab5fe5af"],"text":"A013609 \u2192 basis A013610"},"headline":"A013609 \u2192 basis A013610","chain":null}},{"id":"l-52353753225405d2","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-b289b6e9a15ed6af","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b289b6e9a15ed6af","other":{"id":"n-b289b6e9a15ed6af","name":"basis_map_A007318 of Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["12","12","3"]],"card_rows":[["1"],["6","3"],["12","12","3"],["24","36","18","3"]],"row_sums_preview":["1","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f71648c3ccd02d4","steps":["basis A007318"],"edges":["e-ed772f252d15d000"],"text":"A134059 \u2192 basis A007318"},"headline":"A134059 \u2192 basis A007318","chain":null}},{"id":"l-54bb8ac8fc79ec18","kind":"fingerprint_related","node_a_id":"n-a487462ee8626f77","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a487462ee8626f77","other":{"id":"n-a487462ee8626f77","name":"derivative of basis_map_A013609 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["16","32"]],"card_rows":[["4"],["16","32"],["48","192","192"],["128","768","1536","1024"]],"row_sums_preview":["0","4","48","432","3456","25920","186624","1306368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609","D"],"edges":["e-6abeab86de871564","e-2245d1bbe9580461"],"text":"A130123 \u2192 basis A013609 \u2192 D"},"headline":"A130123 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-56780e192b38be0e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-3009a4c9a888f27a","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3009a4c9a888f27a","other":{"id":"n-3009a4c9a888f27a","name":"derivative of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["96","192"]],"card_rows":[["8"],["96","192"],["960","3840","3840"],["8960","53760","107520","71680"]],"row_sums_preview":["0","8","288","8640","241920","6531840","172440576","4483454976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","D"],"edges":["e-6d76bdb8087cd7e0","e-a92541e56c68bfb9"],"text":"A356546 \u2192 basis A013611 \u2192 D"},"headline":"A356546 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-56da376964fe28f7","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-22326e799ba04515","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-22326e799ba04515","other":{"id":"n-22326e799ba04515","name":"basis_map_A013609 of derivative of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","16"]],"card_rows":[["2"],["8","16"],["24","96","96"],["64","384","768","512"]],"row_sums_preview":["0","2","24","216","1728","12960","93312","653184"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","basis A013609"],"edges":["e-6151a42dffed124f","e-40eb809f16c495b8"],"text":"A130123 \u2192 D \u2192 basis A013609"},"headline":"A130123 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-5a783f3f1c96e197","kind":"fingerprint_related","node_a_id":"n-21cae1d19bcff7ea","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-21cae1d19bcff7ea","other":{"id":"n-21cae1d19bcff7ea","name":"reversal of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["5","30","60","40"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1","reverse"],"edges":["e-45645d17dad7cf5c","e-6e7ac8df8b6e43ee"],"text":"A085880 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A085880 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-5b419bc061b78a6e","kind":"fingerprint_related","node_a_id":"n-8f1f80f146f45463","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f1f80f146f45463","other":{"id":"n-8f1f80f146f45463","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["x \u21a6 x/(x+1)"],"edges":["e-19854cf600900932"],"text":"A038208 \u2192 x \u21a6 x/(x+1)"},"headline":"A038208 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-5bef1808b0a4e43f","kind":"fingerprint_related","node_a_id":"n-6a307f0d4f22d81e","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6a307f0d4f22d81e","other":{"id":"n-6a307f0d4f22d81e","name":"shift_x_plus_one of basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["8","4"],["32","32","8"]],"card_rows":[["2"],["8","4"],["32","32","8"],["128","192","96","16"]],"row_sums_preview":["2","12","72","432","2592","15552","93312","559872"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-23f81812d1886d3d","e-714e5c4cfdfade39"],"text":"A028326 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A028326 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5d349395d9e555d8","kind":"fingerprint_related","node_a_id":"n-a67bbce5c2713a1e","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a67bbce5c2713a1e","other":{"id":"n-a67bbce5c2713a1e","name":"basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","8"],["8","32","32"]],"card_rows":[["2"],["4","8"],["8","32","32"],["16","96","192","128"]],"row_sums_preview":["2","12","72","432","2592","15552","93312","559872"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611"],"edges":["e-3deac81ab612cd2e"],"text":"A028326 \u2192 basis A013611"},"headline":"A028326 \u2192 basis A013611","chain":null}},{"id":"l-5d36b80f7acb9741","kind":"fingerprint_related","node_a_id":"n-de403801a492a76b","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-de403801a492a76b","other":{"id":"n-de403801a492a76b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"]],"card_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"],["1048576","2621440","2621440","1310720","327680","32768"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609"],"edges":["e-6efc8fa5932928ce"],"text":"A038330 \u2192 basis A013609"},"headline":"A038330 \u2192 basis A013609","chain":null}},{"id":"l-60a51796ce808730","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-900c1d03ececa54d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-900c1d03ececa54d","other":{"id":"n-900c1d03ececa54d","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","36"],["324","1296","1296"]],"card_rows":[["1"],["18","36"],["324","1296","1296"],["5832","34992","69984","46656"]],"row_sums_preview":["1","54","2916","157464","8503056","459165024","24794911296","1338925209984"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","basis A013610"],"edges":["e-fc7ebcb9132c171a","e-36efa36d2f73418c"],"text":"A038221 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038221 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-6140ad7bd3e4aa26","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-9f0b2634da8b3f8f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9f0b2634da8b3f8f","other":{"id":"n-9f0b2634da8b3f8f","name":"basis_map_A013610 of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","18"],["81","324","324"]],"card_rows":[["1"],["9","18"],["81","324","324"],["729","4374","8748","5832"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","basis A013610"],"edges":["e-14a3a792ab5fe5af","e-3c8793494a286f3c"],"text":"A013609 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A013609 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-62095fbfc59c16a1","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-045c2e248d2487ed","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-045c2e248d2487ed","other":{"id":"n-045c2e248d2487ed","name":"reversal of basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","24"]],"card_rows":[["1"],["2","4"],["6","24","24"],["24","144","288","192"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["\u03b8+1","basis A013609"],"edges":["e-4dc8adfab569fa5c","e-65bf8c0d0da55b20"],"text":"A136572 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"A136572 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-62bd6d0565776fdd","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2352aed893dd9749","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2352aed893dd9749","other":{"id":"n-2352aed893dd9749","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["24","96","96"]],"card_rows":[["1"],["6","12"],["24","96","96"],["80","480","960","640"]],"row_sums_preview":["1","18","216","2160","19440","163296","1306368","10077696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["basis A013611"],"edges":["e-89b0bff9927fbc5c"],"text":"A094305 \u2192 basis A013611"},"headline":"A094305 \u2192 basis A013611","chain":null}},{"id":"l-62e9c7078f4207c1","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ada652e4c7cffb11","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ada652e4c7cffb11","other":{"id":"n-ada652e4c7cffb11","name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["27","54"],["243","972","972"]],"card_rows":[["3"],["27","54"],["243","972","972"],["2187","13122","26244","17496"]],"row_sums_preview":["3","81","2187","59049","1594323","43046721","1162261467","31381059609"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","basis A013610"],"edges":["e-284e6c48cfe1aa93","e-0afa48aeb2303338"],"text":"A120909 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A120909 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-630b8f53e6d0fcc4","kind":"fingerprint_related","node_a_id":"n-e8ae0bac7c0efceb","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e8ae0bac7c0efceb","other":{"id":"n-e8ae0bac7c0efceb","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["2","8","8"]],"card_rows":[["1"],["2","4"],["2","8","8"],["4","24","48","32"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x/(x+1)"],"edges":["e-9f3b1e4aefe8044e"],"text":"A135838 \u2192 x \u21a6 x/(x+1)"},"headline":"A135838 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-643080df05b75f52","kind":"fingerprint_related","node_a_id":"n-a4d5926477bcae59","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a4d5926477bcae59","other":{"id":"n-a4d5926477bcae59","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["36","18"]],"card_rows":[["3"],["36","18"],["324","324","81"],["2592","3888","1944","324"]],"row_sums_preview":["0","3","54","729","8748","98415","1062882","11160261"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["x \u21a6 x+1","D"],"edges":["e-02e8db4659f506e0","e-9f9c890dbe978eab"],"text":"A038221 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038221 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-658dda7c4f648744","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-4e2b2d6cd8e0ce29","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4e2b2d6cd8e0ce29","other":{"id":"n-4e2b2d6cd8e0ce29","name":"basis_map_A013610 of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","36","36"],["27","162","324","216"],["81","648","1944","2592","1296"]],"card_rows":[["9","36","36"],["27","162","324","216"],["81","648","1944","2592","1296"],["243","2430","9720","19440","19440","7776"]],"row_sums_preview":["81","729","6561","59049","531441","4782969","43046721","387420489"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","basis A013610"],"edges":["e-7bee75fc05394bb8","e-07476b3bf4db265b"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A013610"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-6707b6dbf30ebb8b","kind":"fingerprint_related","node_a_id":"n-0d4899845526eba3","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0d4899845526eba3","other":{"id":"n-0d4899845526eba3","name":"shift_x_plus_one of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["4","4","1"]],"card_rows":[["1"],["-2","-1"],["4","4","1"],["-8","-12","-6","-1"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-4c344817bd191837","e-350089fd6b0d9a4e"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-677dac8db6e9b3fc","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-20043c35aa371b14","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-20043c35aa371b14","other":{"id":"n-20043c35aa371b14","name":"basis_map_A013611 of derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","16"]],"card_rows":[["1"],["8","16"],["60","240","240"],["448","2688","5376","3584"]],"row_sums_preview":["0","1","24","540","12096","272160","6158592","140107968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D","basis A013611"],"edges":["e-592ee9fddf6f118c","e-2c50c41031c27504"],"text":"A085880 \u2192 D \u2192 basis A013611"},"headline":"A085880 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-67bebe516b4a4c10","kind":"fingerprint_related","node_a_id":"n-416d26cbd480eee5","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-416d26cbd480eee5","other":{"id":"n-416d26cbd480eee5","name":"shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","2"],["8","8","2"]],"card_rows":[["2"],["4","2"],["8","8","2"],["16","24","12","2"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318"],"edges":["e-014dd8a0f9594ad5"],"text":"A028326 \u2192 basis A007318"},"headline":"A028326 \u2192 basis A007318","chain":null}},{"id":"l-6903002fa8904267","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ffa298ba6eae841f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ffa298ba6eae841f","other":{"id":"n-ffa298ba6eae841f","name":"basis_map_A013609 of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["12","6"],["48","48","12"]],"card_rows":[["3"],["12","6"],["48","48","12"],["192","288","144","24"]],"row_sums_preview":["3","18","108","648","3888","23328","139968","839808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A007318","basis A013609"],"edges":["e-18f137fab6db112d","e-088d4004b859bbbb"],"text":"A120910 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A120910 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-6913f1868d9f90b0","kind":"fingerprint_related","node_a_id":"n-3f5f8a84df712aee","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3f5f8a84df712aee","other":{"id":"n-3f5f8a84df712aee","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["8","8","2"]],"card_rows":[["1"],["4","2"],["8","8","2"],["32","48","24","4"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x+1"],"edges":["e-7cade0dc9a919c2f"],"text":"A135838 \u2192 x \u21a6 x+1"},"headline":"A135838 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6964087c8b4d0cd8","kind":"fingerprint_related","node_a_id":"n-bb8edef2f9231593","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bb8edef2f9231593","other":{"id":"n-bb8edef2f9231593","name":"derivative of basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-2"],["4","8"]],"card_rows":[["-2"],["4","8"],["-6","-24","-24"],["8","48","96","64"]],"row_sums_preview":["0","-2","12","-54","216","-810","2916","-10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609","D"],"edges":["e-13dcb978bba71940","e-041e311ae133eb6d"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 D"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-6b889223415f1dc8","kind":"fingerprint_related","node_a_id":"n-a76db9b78dbb5551","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a76db9b78dbb5551","other":{"id":"n-a76db9b78dbb5551","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["84","336","336"]],"card_rows":[["1"],["12","24"],["84","336","336"],["448","2688","5376","3584"]],"row_sums_preview":["1","36","756","12096","163296","1959552","21555072","221709312"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611"],"edges":["e-377b2cf400ddb478"],"text":"A178822 \u2192 basis A013611"},"headline":"A178822 \u2192 basis A013611","chain":null}},{"id":"l-70ed376456d45ff1","kind":"fingerprint_related","node_a_id":"n-6ba906f5d673fec6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6ba906f5d673fec6","other":{"id":"n-6ba906f5d673fec6","name":"mobius_x_over_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","4"],["2","8","8"]],"card_rows":[["2"],["2","4"],["2","8","8"],["2","12","24","16"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["x \u21a6 x/(x+1)"],"edges":["e-71255c0d2ce9c4bc"],"text":"A028326 \u2192 x \u21a6 x/(x+1)"},"headline":"A028326 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-72bfac3136fb3e9f","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-69e147a1d9b34e2b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-69e147a1d9b34e2b","other":{"id":"n-69e147a1d9b34e2b","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","128"],["6144","6144","1536"],["131072","196608","98304","16384"]],"card_rows":[["256","128"],["6144","6144","1536"],["131072","196608","98304","16384"],["2621440","5242880","3932160","1310720","163840"]],"row_sums_preview":["384","13824","442368","13271040","382205952","10701766656","293534171136","7925422620672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","D"],"edges":["e-6efc8fa5932928ce","e-4ef7d9becf2e58bf"],"text":"A038330 \u2192 basis A013609 \u2192 D"},"headline":"A038330 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-72fa84547ac6d7d3","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-fe336889ad341afa","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fe336889ad341afa","other":{"id":"n-fe336889ad341afa","name":"basis_map_A013611 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","8"]],"card_rows":[["1"],["4","8"],["12","48","48"],["32","192","384","256"]],"row_sums_preview":["0","1","12","108","864","6480","46656","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-308cf60736924b4c","steps":["basis A013611"],"edges":["e-46f85574b3b04dab"],"text":"A216973 \u2192 basis A013611"},"headline":"A216973 \u2192 basis A013611","chain":null}},{"id":"l-73e1130031845370","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-24f10ea37a6af3ac","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-24f10ea37a6af3ac","other":{"id":"n-24f10ea37a6af3ac","name":"basis_map_A013611 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["18","36"],["108","432","432"]],"card_rows":[["3"],["18","36"],["108","432","432"],["648","3888","7776","5184"]],"row_sums_preview":["3","54","972","17496","314928","5668704","102036672","1836660096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013610","basis A013611"],"edges":["e-3ea97f7a94318269","e-151a98dda3111ffc"],"text":"A120910 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A120910 \u2192 basis A013610 \u2192 basis A013611","chain":null}},{"id":"l-752a9fbde0b74ed9","kind":"fingerprint_related","node_a_id":"n-465e4344a8d94925","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-465e4344a8d94925","other":{"id":"n-465e4344a8d94925","name":"basis_map_A013609 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609"],"edges":["e-6abeab86de871564"],"text":"A130123 \u2192 basis A013609"},"headline":"A130123 \u2192 basis A013609","chain":null}},{"id":"l-75884445adefe678","kind":"fingerprint_related","node_a_id":"n-7933941f4c6059fa","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7933941f4c6059fa","other":{"id":"n-7933941f4c6059fa","name":"reversal of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["6","24","24"]],"card_rows":[["1"],["3","6"],["6","24","24"],["10","60","120","80"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","reverse"],"edges":["e-34e3d9a08d46fb54","e-c543a21d831b0926"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-763d446b57302193","kind":"fingerprint_related","node_a_id":"n-30929436879bea41","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-30929436879bea41","other":{"id":"n-30929436879bea41","name":"shift_x_plus_one of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["40","60","30","5"]],"row_sums_preview":["1","3","27","135","891","5103","31347","185895"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["x \u21a6 x+1"],"edges":["e-ad53dd9247bdcd6e"],"text":"A124860 \u2192 x \u21a6 x+1"},"headline":"A124860 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-78806f190196276b","kind":"fingerprint_related","node_a_id":"n-0df4e06696e956d6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0df4e06696e956d6","other":{"id":"n-0df4e06696e956d6","name":"mobius_x_over_x_plus_one of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","0","0","1","2"],["0","0","0","0","0","0","1","4","4"]],"card_rows":[["0","0","1"],["0","0","0","0","1","2"],["0","0","0","0","0","0","1","4","4"],["0","0","0","0","0","0","0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a8bf9bc375c66ce8","steps":["x \u21a6 x/(x+1)"],"edges":["e-b19d9eb8ab4c3c85"],"text":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)"},"headline":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7c12e04a1031a9d3","kind":"fingerprint_related","node_a_id":"n-b4b87865f0ed6035","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b4b87865f0ed6035","other":{"id":"n-b4b87865f0ed6035","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["120","720","1440","960"]],"row_sums_preview":["1","6","108","3240","136080","7348320","484989120","37829151360"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611"],"edges":["e-20dcbad557ba463b"],"text":"A085881 \u2192 basis A013611"},"headline":"A085881 \u2192 basis A013611","chain":null}},{"id":"l-7d3159361c623d76","kind":"fingerprint_related","node_a_id":"n-c201d29a62400440","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c201d29a62400440","other":{"id":"n-c201d29a62400440","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"]],"card_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"],["7962624","79626240","318504960","637009920","637009920","254803968"]],"row_sums_preview":["5184","373248","26873856","1934917632","139314069504","10030613004288","722204136308736","51998697814228992"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013611"],"edges":["e-c462762e7768ae01"],"text":"A038338 \u2192 basis A013611"},"headline":"A038338 \u2192 basis A013611","chain":null}},{"id":"l-7d5b597643c49f45","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-d687177931d5f117","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-d687177931d5f117","other":{"id":"n-d687177931d5f117","name":"derivative of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["12","6"]],"card_rows":[["1"],["12","6"],["180","180","45"],["3360","5040","2520","420"]],"row_sums_preview":["0","1","18","405","11340","382725","15155910","689593905"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","D"],"edges":["e-5048ce3ebd740a6f","e-6c2800608063eafc"],"text":"A085881 \u2192 basis A007318 \u2192 D"},"headline":"A085881 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-7d82b31bdd88cca2","kind":"fingerprint_related","node_a_id":"n-e621a5e3fc665a4e","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e621a5e3fc665a4e","other":{"id":"n-e621a5e3fc665a4e","name":"mobius_x_over_x_plus_one of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["3","18","36","24"]],"row_sums_preview":["1","3","18","81","405","1944","9477","45927"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["x \u21a6 x/(x+1)"],"edges":["e-ceeda1a209bfa43f"],"text":"basis-map coefficients of A127647 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A127647 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7d9d4d36e7f2c74c","kind":"fingerprint_related","node_a_id":"n-b061091a2d541bea","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b061091a2d541bea","other":{"id":"n-b061091a2d541bea","name":"mobius_x_over_x_plus_one of Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["3","12","12"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["1","6","27","108","405","1458","5103","17496"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["x \u21a6 x/(x+1)"],"edges":["e-bbc53a65557a79f5"],"text":"A003506 \u2192 x \u21a6 x/(x+1)"},"headline":"A003506 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7e374206a7fec776","kind":"fingerprint_related","node_a_id":"n-5317d1468ab69371","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5317d1468ab69371","other":{"id":"n-5317d1468ab69371","name":"basis_map_A013611 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","60"],["180","720","720"]],"card_rows":[["3"],["30","60"],["180","720","720"],["840","5040","10080","6720"]],"row_sums_preview":["3","90","1620","22680","272160","2939328","29393280","277136640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013611"],"edges":["e-6d4f84bcbae4553b"],"text":"A344565 \u2192 basis A013611"},"headline":"A344565 \u2192 basis A013611","chain":null}},{"id":"l-7e6b460b294389ce","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-252c73401458f7fc","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-252c73401458f7fc","other":{"id":"n-252c73401458f7fc","name":"basis_map_A013609 of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-8"],["9","36","36"]],"card_rows":[["1"],["-4","-8"],["9","36","36"],["-16","-96","-192","-128"]],"row_sums_preview":["1","-12","81","-432","2025","-8748","35721","-139968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-23b8fb76a1e0b114","e-65e847d21bf90655"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-7efcf47001c7ceaa","kind":"fingerprint_related","node_a_id":"n-8f0d536701873284","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f0d536701873284","other":{"id":"n-8f0d536701873284","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["8","32","32"]],"card_rows":[["1"],["4","8"],["8","32","32"],["32","192","384","256"]],"row_sums_preview":["1","12","72","864","5184","62208","373248","4478976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013611"],"edges":["e-0f9b3c3168d8e647"],"text":"A135838 \u2192 basis A013611"},"headline":"A135838 \u2192 basis A013611","chain":null}},{"id":"l-80bdc164ab67ed44","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-72bb2eae3768e77e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-72bb2eae3768e77e","other":{"id":"n-72bb2eae3768e77e","name":"basis_map_A013611 of Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["80","480","960","640"]],"row_sums_preview":["1","6","108","2160","45360","979776","21555072","480370176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9a3e4fcd8c6a0286","steps":["basis A013611"],"edges":["e-33fe1e3c92a3e49d"],"text":"A258758 \u2192 basis A013611"},"headline":"A258758 \u2192 basis A013611","chain":null}},{"id":"l-80f261dc0b49fed3","kind":"fingerprint_related","node_a_id":"n-4c8e9ad9dc69215d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4c8e9ad9dc69215d","other":{"id":"n-4c8e9ad9dc69215d","name":"mobius_x_over_x_plus_one of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","24"]],"card_rows":[["1"],["2","4"],["6","24","24"],["24","144","288","192"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["x \u21a6 x/(x+1)"],"edges":["e-dd065961f976b7a9"],"text":"A156992 \u2192 x \u21a6 x/(x+1)"},"headline":"A156992 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-81d587b98d83386d","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-53f60fe904850b39","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-53f60fe904850b39","other":{"id":"n-53f60fe904850b39","name":"basis_map_A013609 of Right shift operator generating 1's in shifted spaces.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","4","4"]],"row_sums_preview":["1","1","3","9","27","81","243","729"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d2ea2ff0a779ba6e","steps":["basis A013609"],"edges":["e-c9edb404dc0a2d79"],"text":"A129186 \u2192 basis A013609"},"headline":"A129186 \u2192 basis A013609","chain":null}},{"id":"l-8520fa89022f3c77","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-f9af849219405014","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f9af849219405014","other":{"id":"n-f9af849219405014","name":"basis_map_A013610 of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["72","288","288"]],"card_rows":[["1"],["6","12"],["72","288","288"],["1296","7776","15552","10368"]],"row_sums_preview":["1","18","648","34992","2519424","226748160","24488801280","3085588961280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611","basis A013610"],"edges":["e-b8ab5069d7db191f","e-b374508ccd824ae5"],"text":"A196347 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A196347 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-87c94443888b0d85","kind":"fingerprint_related","node_a_id":"n-f8c9dbb903f44fdc","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f8c9dbb903f44fdc","other":{"id":"n-f8c9dbb903f44fdc","name":"shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["4","4","1"]],"card_rows":[["1"],["-2","-1"],["4","4","1"],["-8","-12","-6","-1"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1"],"edges":["e-51a320c1f4e00092"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8930617be49972c5","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-3c2e71727d16a914","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3c2e71727d16a914","other":{"id":"n-3c2e71727d16a914","name":"basis_map_A013609 of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["9","36","36"]],"card_rows":[["1"],["4","8"],["9","36","36"],["16","96","192","128"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-3411eba2e020f0d3","e-199f01065e9ef8ba"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-89e23f61e8e3c79b","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-f4b4db1dfdb62066","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f4b4db1dfdb62066","other":{"id":"n-f4b4db1dfdb62066","name":"derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["24"],["336","672"]],"card_rows":[["24"],["336","672"],["2688","10752","10752"],["16128","96768","193536","129024"]],"row_sums_preview":["0","24","1008","24192","435456","6531840","86220288","1034643456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611","D"],"edges":["e-377b2cf400ddb478","e-5e78d90f4dffa0c0"],"text":"A178822 \u2192 basis A013611 \u2192 D"},"headline":"A178822 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-8a8240b00f99e049","kind":"fingerprint_related","node_a_id":"n-17166a4ad91393c8","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-17166a4ad91393c8","other":{"id":"n-17166a4ad91393c8","name":"basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["48","72","36","6"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["x \u21a6 x+1"],"edges":["e-0b89a3d3e0cfd72c"],"text":"A196347 \u2192 x \u21a6 x+1"},"headline":"A196347 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8a99a719f21aeefe","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-46756cb125726a43","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-46756cb125726a43","other":{"id":"n-46756cb125726a43","name":"basis_map_A013609 of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","8"]],"card_rows":[["1"],["4","8"],["18","72","72"],["96","576","1152","768"]],"row_sums_preview":["0","1","12","162","2592","48600","1049760","25719120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","basis A013609"],"edges":["e-789a32df93c7cdb5","e-be69cae2c601f44e"],"text":"A136572 \u2192 D \u2192 basis A013609"},"headline":"A136572 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-8bc9e3c444900e25","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-01adf842a5cd4b2f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-01adf842a5cd4b2f","other":{"id":"n-01adf842a5cd4b2f","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["18"],["108","216"]],"card_rows":[["18"],["108","216"],["486","1944","1944"],["1944","11664","23328","15552"]],"row_sums_preview":["0","18","324","4374","52488","590490","6377292","66961566"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","D"],"edges":["e-284e6c48cfe1aa93","e-091e779e2890bbf3"],"text":"A120909 \u2192 basis A013610 \u2192 D"},"headline":"A120909 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-8c68c5fd043fb43c","kind":"fingerprint_related","node_a_id":"n-a116f96e8f97e756","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a116f96e8f97e756","other":{"id":"n-a116f96e8f97e756","name":"mobius_x_over_x_plus_one of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","1","2"],["0","0","0","0","1","4","4"]],"card_rows":[["1"],["0","0","1","2"],["0","0","0","0","1","4","4"],["0","0","0","0","0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-11b174a1f5e78a68","steps":["x \u21a6 x/(x+1)"],"edges":["e-e69f72c7ed51c84d"],"text":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)"},"headline":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8d605273da2ab1ea","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-9418caaca51598b1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9418caaca51598b1","other":{"id":"n-9418caaca51598b1","name":"basis_map_A013609 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","4"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["0","1","6","27","108","405","1458","5103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["basis A013609"],"edges":["e-10c4ba0bb6f05334"],"text":"A132440 \u2192 basis A013609"},"headline":"A132440 \u2192 basis A013609","chain":null}},{"id":"l-910234396d14cbd1","kind":"fingerprint_related","node_a_id":"n-0cf41fd0b9103ec6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0cf41fd0b9103ec6","other":{"id":"n-0cf41fd0b9103ec6","name":"shift_x_plus_one of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["4","2"]],"card_rows":[["-1"],["4","2"],["-12","-12","-3"],["32","48","24","4"]],"row_sums_preview":["0","-1","6","-27","108","-405","1458","-5103"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D","basis A007318"],"edges":["e-3c99f80a26484c6d","e-420bffb5d93b204a"],"text":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A007318"},"headline":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A007318","chain":null}},{"id":"l-91cac52eae7622a0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-0f3362f24f0fc07b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0f3362f24f0fc07b","other":{"id":"n-0f3362f24f0fc07b","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["44","22"],["1936","1936","484"]],"card_rows":[["1"],["44","22"],["1936","1936","484"],["85184","127776","63888","10648"]],"row_sums_preview":["1","66","4356","287496","18974736","1252332576","82653950016","5455160701056"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013609","basis A007318"],"edges":["e-290484f08292dc0a","e-5c390aa269503e17"],"text":"A038325 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038325 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-92bacd9e8e1eb8c9","kind":"fingerprint_related","node_a_id":"n-24e260a0c0fb324c","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-24e260a0c0fb324c","other":{"id":"n-24e260a0c0fb324c","name":"reversal of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","24"]],"card_rows":[["1"],["2","4"],["6","24","24"],["20","120","240","160"]],"row_sums_preview":["1","6","54","540","5670","61236","673596","7505784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","reverse"],"edges":["e-0f8327ed3206c9f7","e-610623b6dc347e01"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-92e7a3cbc518ce77","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-774ed3fb4827ac2b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-774ed3fb4827ac2b","other":{"id":"n-774ed3fb4827ac2b","name":"derivative of basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","4"]],"card_rows":[["1"],["8","4"],["72","72","18"],["768","1152","576","96"]],"row_sums_preview":["0","1","12","162","2592","48600","1049760","25719120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["x \u21a6 x+1","D"],"edges":["e-0b89a3d3e0cfd72c","e-c7e22bf2b58e3bf0"],"text":"A196347 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A196347 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-956eecf543ae1a70","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-df93d57821958126","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-df93d57821958126","other":{"id":"n-df93d57821958126","name":"basis_map_A013611 of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","48"],["336","1344","1344"]],"card_rows":[["1"],["24","48"],["336","1344","1344"],["3584","21504","43008","28672"]],"row_sums_preview":["1","72","3024","96768","2612736","62705664","1379524608","28378791936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609","basis A013611"],"edges":["e-5270021ea4ce6d61","e-c28347bded7fc435"],"text":"A178822 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A178822 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-95b38d68d33191e0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-e7fd062b528404dc","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e7fd062b528404dc","other":{"id":"n-e7fd062b528404dc","name":"basis_map_A013611 of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["4","16","16"]],"card_rows":[["1"],["-2","-4"],["4","16","16"],["-8","-48","-96","-64"]],"row_sums_preview":["1","-6","36","-216","1296","-7776","46656","-279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318","basis A013611"],"edges":["e-4c344817bd191837","e-d77cbc7443516c7d"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 basis A013611","chain":null}},{"id":"l-9850052f2e15a0d3","kind":"fingerprint_related","node_a_id":"n-66e925f07bda356b","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-66e925f07bda356b","other":{"id":"n-66e925f07bda356b","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","10"],["15","60","60"]],"card_rows":[["1"],["5","10"],["15","60","60"],["35","210","420","280"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318","reverse"],"edges":["e-484d449f4d0f99c5","e-0b3de33668e94b9b"],"text":"A178821 \u2192 basis A007318 \u2192 reverse"},"headline":"A178821 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-9a089c7ba51f1170","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-e3c2e6e725bdc109","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e3c2e6e725bdc109","other":{"id":"n-e3c2e6e725bdc109","name":"basis_map_A013611 of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["56","336","672","448"]],"row_sums_preview":["1","6","108","1512","24624","396576","6578496","110014848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["basis A013611"],"edges":["e-42d2bb17aed20ab4"],"text":"A376827 \u2192 basis A013611"},"headline":"A376827 \u2192 basis A013611","chain":null}},{"id":"l-9aeacd0008c47763","kind":"fingerprint_related","node_a_id":"n-8f9866a4e131e95d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f9866a4e131e95d","other":{"id":"n-8f9866a4e131e95d","name":"basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","11"],["484","484","121"]],"card_rows":[["1"],["22","11"],["484","484","121"],["10648","15972","7986","1331"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318"],"edges":["e-a5d0659f5315d3b7"],"text":"A038325 \u2192 basis A007318"},"headline":"A038325 \u2192 basis A007318","chain":null}},{"id":"l-9af6700f86f8e178","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-abd5c0abf48a5f04","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-abd5c0abf48a5f04","other":{"id":"n-abd5c0abf48a5f04","name":"basis_map_A007318 of Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","5"],["20","20","5"]],"card_rows":[["1"],["10","5"],["20","20","5"],["40","60","30","5"]],"row_sums_preview":["1","15","45","135","405","1215","3645","10935"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f0857c877d3df8d4","steps":["basis A007318"],"edges":["e-c89509677a80f66d"],"text":"A135089 \u2192 basis A007318"},"headline":"A135089 \u2192 basis A007318","chain":null}},{"id":"l-9b329f8302195109","kind":"fingerprint_related","node_a_id":"n-f5424c3aafc4063f","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f5424c3aafc4063f","other":{"id":"n-f5424c3aafc4063f","name":"shift_x_plus_one of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["28","28","7"]],"card_rows":[["1"],["8","4"],["28","28","7"],["80","120","60","10"]],"row_sums_preview":["1","12","63","270","1053","3888","13851","48114"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-28824f426538156a","e-7344680469994f19"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-9baacd94d71b7a62","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-20e826b0ccececf7","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-20e826b0ccececf7","other":{"id":"n-20e826b0ccececf7","name":"mobius_x_over_x_plus_one of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","22"],["121","484","484"]],"card_rows":[["1"],["11","22"],["121","484","484"],["1331","7986","15972","10648"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["x \u21a6 x/(x+1)"],"edges":["e-efe1851c24f1d557"],"text":"A038325 \u2192 x \u21a6 x/(x+1)"},"headline":"A038325 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9c8c2f7c789b2c8e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-8dd53223cceac8df","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8dd53223cceac8df","other":{"id":"n-8dd53223cceac8df","name":"basis_map_A013611 of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["24","144","288","192"]],"row_sums_preview":["1","6","72","648","6480","62208","606528","5878656"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["basis A013611"],"edges":["e-7a202e1e5df42599"],"text":"basis-map coefficients of A127647 \u2192 basis A013611"},"headline":"basis-map coefficients of A127647 \u2192 basis A013611","chain":null}},{"id":"l-9d527abdb2beaf74","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-7af0d18f411a3a80","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7af0d18f411a3a80","other":{"id":"n-7af0d18f411a3a80","name":"basis_map_A013609 of theta_plus_one_squared of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["36","144","144"]],"card_rows":[["1"],["8","16"],["36","144","144"],["128","768","1536","1024"]],"row_sums_preview":["1","24","324","3456","32400","279936","2286144","17915904"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["basis A013611"],"edges":["e-586f35eff31e86f6"],"text":"A135065 \u2192 basis A013611"},"headline":"A135065 \u2192 basis A013611","chain":null}},{"id":"l-9d99dd8fcfd3cb91","kind":"fingerprint_related","node_a_id":"n-1497736550841122","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1497736550841122","other":{"id":"n-1497736550841122","name":"reversal of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["9","36","36"]],"card_rows":[["1"],["4","8"],["9","36","36"],["16","96","192","128"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","reverse"],"edges":["e-18ace392e620d7bd","e-09f10e821d5e35b0"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-a05c9c0b6a54de80","kind":"fingerprint_related","node_a_id":"n-0793a8fd2b196f0d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0793a8fd2b196f0d","other":{"id":"n-0793a8fd2b196f0d","name":"basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609"],"edges":["e-66f8525b021adc45"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609","chain":null}},{"id":"l-a1304ea057419ab0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-acb5f60cda3bf411","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-acb5f60cda3bf411","other":{"id":"n-acb5f60cda3bf411","name":"basis_map_A013610 of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","9"],["324","324","81"]],"card_rows":[["1"],["18","9"],["324","324","81"],["5832","8748","4374","729"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","basis A013610"],"edges":["e-011a72a323f5e83e","e-4c2b72166ade0baa"],"text":"A147716 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A147716 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-a24cf3a5bec1a251","kind":"fingerprint_related","node_a_id":"n-eb97e93fd65cf0d3","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-eb97e93fd65cf0d3","other":{"id":"n-eb97e93fd65cf0d3","name":"shift_x_plus_one of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["56","84","42","7"]],"row_sums_preview":["1","3","27","189","1539","12393","102789","859491"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["basis A007318"],"edges":["e-9e3965a70175b87c"],"text":"A376827 \u2192 basis A007318"},"headline":"A376827 \u2192 basis A007318","chain":null}},{"id":"l-a48bbe6562d332de","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-1410900940bde40c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1410900940bde40c","other":{"id":"n-1410900940bde40c","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["36","72"],["756","3024","3024"]],"card_rows":[["1"],["36","72"],["756","3024","3024"],["12096","72576","145152","96768"]],"row_sums_preview":["1","108","6804","326592","13226976","476171136","15713647488","484878265344"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611","basis A013610"],"edges":["e-377b2cf400ddb478","e-8c8ed9ed09147558"],"text":"A178822 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-a4c40b8b68832f6d","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-c3c16d1eee30ac4f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c3c16d1eee30ac4f","other":{"id":"n-c3c16d1eee30ac4f","name":"derivative of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["32","64"]],"card_rows":[["4"],["32","64"],["240","960","960"],["1792","10752","21504","14336"]],"row_sums_preview":["0","4","96","2160","48384","1088640","24634368","560431872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611","D"],"edges":["e-9f3d1c5f0f510549","e-3878470d6a9d0fac"],"text":"A085880 \u2192 basis A013611 \u2192 D"},"headline":"A085880 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-a5dbab6026dae75c","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-7a7b1ea2b00dadf1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7a7b1ea2b00dadf1","other":{"id":"n-7a7b1ea2b00dadf1","name":"basis_map_A013611 of basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-8"],["16","64","64"]],"card_rows":[["1"],["-4","-8"],["16","64","64"],["-64","-384","-768","-512"]],"row_sums_preview":["1","-12","144","-1728","20736","-248832","2985984","-35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013609","basis A013611"],"edges":["e-b6cdc8ab0f2a09ef","e-0246c3e671fb3d48"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-a83819cabf2ed87b","kind":"fingerprint_related","node_a_id":"n-63d7f3b1eb8cf1f2","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-63d7f3b1eb8cf1f2","other":{"id":"n-63d7f3b1eb8cf1f2","name":"shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["36","36","9"]],"card_rows":[["1"],["8","4"],["36","36","9"],["128","192","96","16"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1"],"edges":["e-18ace392e620d7bd"],"text":"A135065 \u2192 x \u21a6 x+1"},"headline":"A135065 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a8e21cf9912bf101","kind":"fingerprint_related","node_a_id":"n-5a242687ab527719","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5a242687ab527719","other":{"id":"n-5a242687ab527719","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["84","84","21"]],"card_rows":[["1"],["12","6"],["84","84","21"],["448","672","336","56"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1"],"edges":["e-a52727d7b51dd683"],"text":"A178822 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a99cf489709e876c","kind":"fingerprint_related","node_a_id":"n-35019173dc21748d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-35019173dc21748d","other":{"id":"n-35019173dc21748d","name":"shift_x_plus_one of shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","4","1"],["8","12","6","1"],["16","32","24","8","1"]],"card_rows":[["4","4","1"],["8","12","6","1"],["16","32","24","8","1"],["32","80","80","40","10","1"]],"row_sums_preview":["9","27","81","243","729","2187","6561","19683"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-cb9b2e4b7b317e85","e-0062e7c419b6ed2c"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-aaece17e8f3887b9","kind":"fingerprint_related","node_a_id":"n-51b4f5e7901c2b50","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-51b4f5e7901c2b50","other":{"id":"n-51b4f5e7901c2b50","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","3"],["12","12","3"]],"card_rows":[["3"],["6","3"],["12","12","3"],["24","36","18","3"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120910"],"derivation":{"root":"n-51b4f5e7901c2b50","steps":[],"edges":[],"text":"A120910"},"headline":"A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","chain":null}},{"id":"l-abe068df67997a03","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ec306f637426be58","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec306f637426be58","other":{"id":"n-ec306f637426be58","name":"basis_map_A013609 of theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-8"],["9","36","36"]],"card_rows":[["1"],["-4","-8"],["9","36","36"],["-16","-96","-192","-128"]],"row_sums_preview":["1","-12","81","-432","2025","-8748","35721","-139968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-d58e809d3242b4fa","e-488371e9d79ed3a5"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-ac26f3b2035f4882","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2e1f9db2c3ee865d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2e1f9db2c3ee865d","other":{"id":"n-2e1f9db2c3ee865d","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","48"],["360","1440","1440"]],"card_rows":[["1"],["24","48"],["360","1440","1440"],["4320","25920","51840","34560"]],"row_sums_preview":["1","72","3240","116640","3674160","105815808","2857026816","73466403840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","basis A013610"],"edges":["e-159425c642d412bb","e-8e4f6926931d1f02"],"text":"A178820 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A178820 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-aed246960bca0bc5","kind":"fingerprint_related","node_a_id":"n-4dab44b954732fbb","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4dab44b954732fbb","other":{"id":"n-4dab44b954732fbb","name":"basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["160","960","1920","1280"]],"row_sums_preview":["1","12","216","4320","90720","1959552","43110144","960740352"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611"],"edges":["e-6d76bdb8087cd7e0"],"text":"A356546 \u2192 basis A013611"},"headline":"A356546 \u2192 basis A013611","chain":null}},{"id":"l-afcbe605019efad0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-400dbfc2a018f20e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-400dbfc2a018f20e","other":{"id":"n-400dbfc2a018f20e","name":"basis_map_A013609 of Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["2","8","8"]],"card_rows":[["1"],["2","4"],["2","8","8"],["2","12","24","16"]],"row_sums_preview":["1","6","18","54","162","486","1458","4374"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-356713338f14325b","steps":["basis A013609"],"edges":["e-ed3550e9180ecf21"],"text":"A136521 \u2192 basis A013609"},"headline":"A136521 \u2192 basis A013609","chain":null}},{"id":"l-afe04d08029481a4","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-86d2e7430fa8b666","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-86d2e7430fa8b666","other":{"id":"n-86d2e7430fa8b666","name":"basis_map_A013609 of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","4"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["0","1","6","27","108","405","1458","5103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","basis A013609"],"edges":["e-198d139196a8bb12","e-16cca84abbc6f972"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-b0dde18b821c1b42","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-d5693ca6982ed0eb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-d5693ca6982ed0eb","other":{"id":"n-d5693ca6982ed0eb","name":"basis_map_A013611 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["16","32"]],"card_rows":[["2"],["16","32"],["96","384","384"],["512","3072","6144","4096"]],"row_sums_preview":["0","2","48","864","13824","207360","2985984","41803776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D","basis A013611"],"edges":["e-a49f8ff31c907f66","e-35774e8b574578f1"],"text":"A038208 \u2192 D \u2192 basis A013611"},"headline":"A038208 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-b16bd76d9ef25121","kind":"fingerprint_related","node_a_id":"n-3b6ec7300891f6df","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3b6ec7300891f6df","other":{"id":"n-3b6ec7300891f6df","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["6","24","24"]],"card_rows":[["1"],["3","6"],["6","24","24"],["10","60","120","80"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x/(x+1)"],"edges":["e-43602683ebc00dc1"],"text":"A094305 \u2192 x \u21a6 x/(x+1)"},"headline":"A094305 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b1d04b007ac11450","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-a0bec09b71e3cb85","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a0bec09b71e3cb85","other":{"id":"n-a0bec09b71e3cb85","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["96","96","24"]],"card_rows":[["1"],["8","4"],["96","96","24"],["1536","2304","1152","192"]],"row_sums_preview":["1","12","216","5184","155520","5598720","235146240","11287019520"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013609","basis A007318"],"edges":["e-b7289e9f518eca68","e-03bc86f7ebb189b6"],"text":"A156992 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A156992 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-b218ed1ccca8ba45","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-9fbed2f13913339a","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9fbed2f13913339a","other":{"id":"n-9fbed2f13913339a","name":"derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["20"],["240","480"]],"card_rows":[["20"],["240","480"],["1680","6720","6720"],["8960","53760","107520","71680"]],"row_sums_preview":["0","20","720","15120","241920","3265920","39191040","431101440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","D"],"edges":["e-4faaf8fb320b6fca","e-d2c6e9bc59c47b9a"],"text":"A178821 \u2192 basis A013611 \u2192 D"},"headline":"A178821 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-b393ba0f0e283276","kind":"fingerprint_related","node_a_id":"n-6e3adcccc1f149b8","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6e3adcccc1f149b8","other":{"id":"n-6e3adcccc1f149b8","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["36","72"]],"card_rows":[["6"],["36","72"],["162","648","648"],["648","3888","7776","5184"]],"row_sums_preview":["0","6","108","1458","17496","196830","2125764","22320522"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","D"],"edges":["e-14a3a792ab5fe5af","e-ffc48828f1203577"],"text":"A013609 \u2192 basis A013610 \u2192 D"},"headline":"A013609 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-b4e44f44095d39ee","kind":"fingerprint_related","node_a_id":"n-c1bb379fae9335fa","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c1bb379fae9335fa","other":{"id":"n-c1bb379fae9335fa","name":"derivative of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","8"],["6","24","24"],["8","48","96","64"]],"card_rows":[["4","8"],["6","24","24"],["8","48","96","64"],["10","80","240","320","160"]],"row_sums_preview":["12","54","216","810","2916","10206","34992","118098"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","D"],"edges":["e-7bee75fc05394bb8","e-f12d91cadd94f3d1"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-b5dbe02502104b5a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5ed6981065d6360f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5ed6981065d6360f","other":{"id":"n-5ed6981065d6360f","name":"basis_map_A013611 of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["40","240","480","320"]],"row_sums_preview":["1","6","108","1080","14256","163296","2006208","23794560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["basis A013611"],"edges":["e-951258f79395820b"],"text":"A124860 \u2192 basis A013611"},"headline":"A124860 \u2192 basis A013611","chain":null}},{"id":"l-b62d63c72f3ab3ab","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5e60b7e4ee0fb197","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5e60b7e4ee0fb197","other":{"id":"n-5e60b7e4ee0fb197","name":"basis_map_A013610 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["9","36","36"]],"card_rows":[["1"],["3","6"],["9","36","36"],["27","162","324","216"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","basis A013610"],"edges":["e-7869781294d5f3a8","e-1f82546f937ff93f"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-b652ec6d7c0985d5","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-e2eb483e26b7616c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e2eb483e26b7616c","other":{"id":"n-e2eb483e26b7616c","name":"basis_map_A007318 of Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["8","8","2"]],"card_rows":[["1"],["4","2"],["8","8","2"],["16","24","12","2"]],"row_sums_preview":["1","6","18","54","162","486","1458","4374"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f70486c726c7ad1c","steps":["basis A007318"],"edges":["e-77aa3ceaad4c52e6"],"text":"A134058 \u2192 basis A007318"},"headline":"A134058 \u2192 basis A007318","chain":null}},{"id":"l-b7f4e9a1a4b18399","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-c5f633c08731c6c1","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c5f633c08731c6c1","other":{"id":"n-c5f633c08731c6c1","name":"basis_map_A013611 of Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","20"],["20","80","80"]],"card_rows":[["1"],["10","20"],["20","80","80"],["40","240","480","320"]],"row_sums_preview":["1","30","180","1080","6480","38880","233280","1399680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f0857c877d3df8d4","steps":["basis A013611"],"edges":["e-a52dedd5f892782b"],"text":"A135089 \u2192 basis A013611"},"headline":"A135089 \u2192 basis A013611","chain":null}},{"id":"l-b8b5dc56da00582d","kind":"fingerprint_related","node_a_id":"n-2676cf706342980f","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2676cf706342980f","other":{"id":"n-2676cf706342980f","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","10"],["240","240","60"]],"card_rows":[["1"],["20","10"],["240","240","60"],["2240","3360","1680","280"]],"row_sums_preview":["1","30","540","7560","90720","979776","9797760","92378880"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","reverse"],"edges":["e-4faaf8fb320b6fca","e-69af5a4627620de3"],"text":"A178821 \u2192 basis A013611 \u2192 reverse"},"headline":"A178821 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-b918451770fea4be","kind":"fingerprint_related","node_a_id":"n-65a2790ea28179de","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-65a2790ea28179de","other":{"id":"n-65a2790ea28179de","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["28","28","7"]],"card_rows":[["1"],["6","3"],["28","28","7"],["120","180","90","15"]],"row_sums_preview":["1","9","63","405","2511","15309","92583","557685"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["x \u21a6 x+1"],"edges":["e-f5c0f2b9bf2022e2"],"text":"A134346 \u2192 x \u21a6 x+1"},"headline":"A134346 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ba6feb7b92c755ad","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-775b9e7cf85328d0","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-775b9e7cf85328d0","other":{"id":"n-775b9e7cf85328d0","name":"basis_map_A013611 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["64","256","256"]],"card_rows":[["1"],["8","16"],["64","256","256"],["512","3072","6144","4096"]],"row_sums_preview":["1","24","576","13824","331776","7962624","191102976","4586471424"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A013611","basis A013611"],"edges":["e-b3c62654af8090af","e-1f6333fa4ac67c30"],"text":"A027465 \u2192 basis A013611 \u2192 basis A013611"},"headline":"A027465 \u2192 basis A013611 \u2192 basis A013611","chain":null}},{"id":"l-bab84826803e5a53","kind":"fingerprint_related","node_a_id":"n-7f05fd9e9c6948d8","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7f05fd9e9c6948d8","other":{"id":"n-7f05fd9e9c6948d8","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["36","36","9"]],"card_rows":[["1"],["6","3"],["36","36","9"],["216","324","162","27"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["x \u21a6 x+1"],"edges":["e-02e8db4659f506e0"],"text":"A038221 \u2192 x \u21a6 x+1"},"headline":"A038221 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-bd34a554bef2860e","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-b0b9f0e48753af20","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b0b9f0e48753af20","other":{"id":"n-b0b9f0e48753af20","name":"basis_map_A013609 of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["27","108","108"]],"card_rows":[["1"],["8","16"],["27","108","108"],["64","384","768","512"]],"row_sums_preview":["1","24","243","1728","10125","52488","250047","1119744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-048aefa516697baf","e-7e466a04f17f31a0"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-be4e3cc9a384c2ec","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-9adeec4a52d3b5a6","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9adeec4a52d3b5a6","other":{"id":"n-9adeec4a52d3b5a6","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["18","72","72"]],"card_rows":[["1"],["3","6"],["18","72","72"],["162","972","1944","1296"]],"row_sums_preview":["1","9","162","4374","157464","7085880","382637520","24106163760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","basis A013610"],"edges":["e-456dc355a6d78eff","e-049a10f8117b30e4"],"text":"A136572 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A136572 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-bfb931dbe7952918","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-130a8064cebe5f50","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-130a8064cebe5f50","other":{"id":"n-130a8064cebe5f50","name":"basis_map_A013609 of theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["3","12","12"]],"card_rows":[["1"],["-2","-4"],["3","12","12"],["-4","-24","-48","-32"]],"row_sums_preview":["1","-6","27","-108","405","-1458","5103","-17496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["\u03b8+1","basis A013609"],"edges":["e-1d47beb63d4711b8","e-ec40b5f07e130932"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-bfed332e586e1e1a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-85a04fa5e3ccdb46","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-85a04fa5e3ccdb46","other":{"id":"n-85a04fa5e3ccdb46","name":"basis_map_A007318 of Numbers in (4,4)-Pascal triangle .","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["16","16","4"]],"card_rows":[["1"],["8","4"],["16","16","4"],["32","48","24","4"]],"row_sums_preview":["1","12","36","108","324","972","2916","8748"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3aa21f78720c1d0c","steps":["basis A007318"],"edges":["e-2c77ff85ae2ce5d6"],"text":"A132200 \u2192 basis A007318"},"headline":"A132200 \u2192 basis A007318","chain":null}},{"id":"l-c0e0e60a30e04d78","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2159e97e57c9c138","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2159e97e57c9c138","other":{"id":"n-2159e97e57c9c138","name":"basis_map_A013611 of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","40"],["240","960","960"]],"card_rows":[["1"],["20","40"],["240","960","960"],["2240","13440","26880","17920"]],"row_sums_preview":["1","60","2160","60480","1451520","31352832","627056640","11824496640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013609","basis A013611"],"edges":["e-7c97406b114f1661","e-a8148f450b06b98d"],"text":"A178821 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A178821 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-c132d8c02abd01b8","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5b0c1f0fe75b55a7","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5b0c1f0fe75b55a7","other":{"id":"n-5b0c1f0fe75b55a7","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","324","81"],["5832","8748","4374","729"],["104976","209952","157464","52488","6561"]],"card_rows":[["324","324","81"],["5832","8748","4374","729"],["104976","209952","157464","52488","6561"],["1889568","4723920","4723920","2361960","590490","59049"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-5083e5230277d126","e-6fd9639eef78df97"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-c1860cae1b6ba771","kind":"fingerprint_related","node_a_id":"n-73bb59c333e132f0","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-73bb59c333e132f0","other":{"id":"n-73bb59c333e132f0","name":"mobius_x_over_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x/(x+1)"],"edges":["e-26d899ee6632b86c"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c1a6910e5d8e32f6","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-ae5152aa1415203e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ae5152aa1415203e","other":{"id":"n-ae5152aa1415203e","name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","12"]],"card_rows":[["1"],["1","2"],["3","12","12"],["15","90","180","120"]],"row_sums_preview":["1","3","27","405","8505","229635","7577955","295540245"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["x \u21a6 x/(x+1)"],"edges":["e-52aefaf6833022a0"],"text":"A085881 \u2192 x \u21a6 x/(x+1)"},"headline":"A085881 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c24934e2703fd0c9","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-64de264fa51c72b9","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-64de264fa51c72b9","other":{"id":"n-64de264fa51c72b9","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","288"],["20736","20736","5184"],["663552","995328","497664","82944"]],"card_rows":[["576","288"],["20736","20736","5184"],["663552","995328","497664","82944"],["19906560","39813120","29859840","9953280","1244160"]],"row_sums_preview":["864","46656","2239488","100776960","4353564672","182849716224","7522959753216","304679870005248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1","D"],"edges":["e-240272ddc3b4d83d","e-f92a2afbd5cad65b"],"text":"A038338 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038338 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-c404f470c52bd7c4","kind":"fingerprint_related","node_a_id":"n-9466c8441e24ce06","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9466c8441e24ce06","other":{"id":"n-9466c8441e24ce06","name":"basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609"],"edges":["e-13dcb978bba71940"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609","chain":null}},{"id":"l-c480d96bbd33bf95","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5fff3b471ddcbe1b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5fff3b471ddcbe1b","other":{"id":"n-5fff3b471ddcbe1b","name":"basis_map_A013610 of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["12","24"],["72","288","288"]],"card_rows":[["2"],["12","24"],["72","288","288"],["432","2592","5184","3456"]],"row_sums_preview":["2","36","648","11664","209952","3779136","68024448","1224440064"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611","basis A013610"],"edges":["e-3deac81ab612cd2e","e-c643c16d102a1362"],"text":"A028326 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A028326 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-c5c1c3896db90da0","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-5ecc40f66e11686c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5ecc40f66e11686c","other":{"id":"n-5ecc40f66e11686c","name":"basis_map_A013611 of basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["8","16"],["32","128","128"]],"card_rows":[["2"],["8","16"],["32","128","128"],["128","768","1536","1024"]],"row_sums_preview":["2","24","288","3456","41472","497664","5971968","71663616"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609","basis A013611"],"edges":["e-23f81812d1886d3d","e-15d4864bdab70165"],"text":"A028326 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A028326 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-c67762f86293d185","kind":"fingerprint_related","node_a_id":"n-8dc7b8a12ecdf1ca","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8dc7b8a12ecdf1ca","other":{"id":"n-8dc7b8a12ecdf1ca","name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["5","30","60","40"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x/(x+1)"],"edges":["e-e1834b9d49e58efd"],"text":"A085880 \u2192 x \u21a6 x/(x+1)"},"headline":"A085880 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c7d12bba4b1a400f","kind":"fingerprint_related","node_a_id":"n-4766ad62f3b3a125","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4766ad62f3b3a125","other":{"id":"n-4766ad62f3b3a125","name":"basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["192","1152","2304","1536"]],"row_sums_preview":["1","12","216","5184","155520","5598720","235146240","11287019520"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611"],"edges":["e-413ab29272bdfecc"],"text":"A156992 \u2192 basis A013611"},"headline":"A156992 \u2192 basis A013611","chain":null}},{"id":"l-c8a272a2ec11cdc0","kind":"fingerprint_related","node_a_id":"n-03ac40ba4fd62e4c","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-03ac40ba4fd62e4c","other":{"id":"n-03ac40ba4fd62e4c","name":"basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["120","180","90","15"]],"row_sums_preview":["1","3","27","405","8505","229635","7577955","295540245"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318"],"edges":["e-5048ce3ebd740a6f"],"text":"A085881 \u2192 basis A007318"},"headline":"A085881 \u2192 basis A007318","chain":null}},{"id":"l-c919ebcbb060aa98","kind":"fingerprint_related","node_a_id":"n-9805b07a338913cb","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9805b07a338913cb","other":{"id":"n-9805b07a338913cb","name":"basis_map_A013611 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","44"],["484","1936","1936"]],"card_rows":[["1"],["22","44"],["484","1936","1936"],["10648","63888","127776","85184"]],"row_sums_preview":["1","66","4356","287496","18974736","1252332576","82653950016","5455160701056"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013611"],"edges":["e-6a2839a9935a683b"],"text":"A038325 \u2192 basis A013611"},"headline":"A038325 \u2192 basis A013611","chain":null}},{"id":"l-cb407d56e932162b","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-67925971be0d4512","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-67925971be0d4512","other":{"id":"n-67925971be0d4512","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["28","112","112"]],"card_rows":[["1"],["6","12"],["28","112","112"],["120","720","1440","960"]],"row_sums_preview":["1","18","252","3240","40176","489888","5925312","71383680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["basis A013611"],"edges":["e-860b28b1d171bb4f"],"text":"A134346 \u2192 basis A013611"},"headline":"A134346 \u2192 basis A013611","chain":null}},{"id":"l-cce51d257a3031af","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-6548bc84b2c67c3c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6548bc84b2c67c3c","other":{"id":"n-6548bc84b2c67c3c","name":"basis_map_A013609 of theta_plus_one of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["12","48","48"]],"card_rows":[["1"],["4","8"],["12","48","48"],["32","192","384","256"]],"row_sums_preview":["1","12","108","864","6480","46656","326592","2239488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["\u03b8+1","basis A013609"],"edges":["e-c327b1a68506ba1f","e-7e9956102cbb42a5"],"text":"A130123 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"A130123 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-ceef601aba8e0567","kind":"fingerprint_related","node_a_id":"n-f69fa95c3f9b63e6","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f69fa95c3f9b63e6","other":{"id":"n-f69fa95c3f9b63e6","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["7","28","28"]],"card_rows":[["1"],["3","6"],["7","28","28"],["15","90","180","120"]],"row_sums_preview":["1","9","63","405","2511","15309","92583","557685"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["x \u21a6 x/(x+1)"],"edges":["e-256311ac4ccfd963"],"text":"A134346 \u2192 x \u21a6 x/(x+1)"},"headline":"A134346 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d14273a538264a70","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-a03980573c4a5b72","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a03980573c4a5b72","other":{"id":"n-a03980573c4a5b72","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["144","144","36"]],"card_rows":[["1"],["12","6"],["144","144","36"],["1728","2592","1296","216"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","basis A013610"],"edges":["e-88ca34e9877f4f47","e-010b0c63b8bd0785"],"text":"A038291 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038291 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-d229e3b40fe2502e","kind":"fingerprint_related","node_a_id":"n-989caf4cbde8ca55","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-989caf4cbde8ca55","other":{"id":"n-989caf4cbde8ca55","name":"shift_x_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["16","8"]],"card_rows":[["2"],["16","8"],["96","96","24"],["512","768","384","64"]],"row_sums_preview":["0","2","24","216","1728","12960","93312","653184"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318","D"],"edges":["e-1756f066a00732e3","e-2a5f625c70a7ffa5"],"text":"A038208 \u2192 basis A007318 \u2192 D"},"headline":"A038208 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-d2f2239ba257520a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-9227fcfb7651bd0b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9227fcfb7651bd0b","other":{"id":"n-9227fcfb7651bd0b","name":"basis_map_A013611 of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["32","128","128"]],"card_rows":[["1"],["4","8"],["32","128","128"],["320","1920","3840","2560"]],"row_sums_preview":["1","12","288","8640","290304","10450944","394149888","15371845632"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609","basis A013611"],"edges":["e-ccc293aee69469d4","e-59520e383e9e4b81"],"text":"A085880 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A085880 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-d40e6e07fd0fa074","kind":"fingerprint_related","node_a_id":"n-e69b830457b341cf","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e69b830457b341cf","other":{"id":"n-e69b830457b341cf","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["40","240","480","320"]],"row_sums_preview":["1","6","72","1080","18144","326592","6158592","120092544"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611"],"edges":["e-9f3d1c5f0f510549"],"text":"A085880 \u2192 basis A013611"},"headline":"A085880 \u2192 basis A013611","chain":null}},{"id":"l-d4647556d1dde6e3","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-73f1b843be1a1e11","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-73f1b843be1a1e11","other":{"id":"n-73f1b843be1a1e11","name":"basis_map_A013611 of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["4","8"]],"card_rows":[["-1"],["4","8"],["-12","-48","-48"],["32","192","384","256"]],"row_sums_preview":["0","-1","12","-108","864","-6480","46656","-326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D","basis A013611"],"edges":["e-3c99f80a26484c6d","e-1b0cd95cf5d97491"],"text":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-d4de5dc2988dfe61","kind":"fingerprint_related","node_a_id":"n-9d1e54ec2606728d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9d1e54ec2606728d","other":{"id":"n-9d1e54ec2606728d","name":"shift_x_plus_one of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["4","4","1"]],"card_rows":[["1"],["2","1"],["4","4","1"],["8","12","6","1"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","reverse"],"edges":["e-7869781294d5f3a8","e-0fd29bc52a900dd2"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 reverse"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-d742a6025f205a17","kind":"fingerprint_related","node_a_id":"n-ee1ca1411a9c8a32","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ee1ca1411a9c8a32","other":{"id":"n-ee1ca1411a9c8a32","name":"derivative of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-2"],["4","8"]],"card_rows":[["-2"],["4","8"],["-6","-24","-24"],["8","48","96","64"]],"row_sums_preview":["0","-2","12","-54","216","-810","2916","-10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","D"],"edges":["e-66f8525b021adc45","e-61c4a702093cb9c8"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 D"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-d79ebb45607dccd2","kind":"fingerprint_related","node_a_id":"n-fff9dc1c6c2891b5","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fff9dc1c6c2891b5","other":{"id":"n-fff9dc1c6c2891b5","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["16","16","4"]],"card_rows":[["1"],["4","2"],["16","16","4"],["64","96","48","8"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318"],"edges":["e-1756f066a00732e3"],"text":"A038208 \u2192 basis A007318"},"headline":"A038208 \u2192 basis A007318","chain":null}},{"id":"l-d7b11ccb44269da7","kind":"fingerprint_related","node_a_id":"n-a43ddbc0e3e9110d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a43ddbc0e3e9110d","other":{"id":"n-a43ddbc0e3e9110d","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","8"],["160","160","40"]],"card_rows":[["1"],["16","8"],["160","160","40"],["1280","1920","960","160"]],"row_sums_preview":["1","24","360","4320","45360","435456","3919104","33592320"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","reverse"],"edges":["e-159425c642d412bb","e-41c65371e31fe387"],"text":"A178820 \u2192 basis A013611 \u2192 reverse"},"headline":"A178820 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-d955644db68cee56","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-37b2531d61071c9f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-37b2531d61071c9f","other":{"id":"n-37b2531d61071c9f","name":"basis_map_A013609 of theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["6","24","24"]],"card_rows":[["1"],["-2","-4"],["6","24","24"],["-24","-144","-288","-192"]],"row_sums_preview":["1","-6","54","-648","9720","-174960","3674160","-88179840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1","basis A013609"],"edges":["e-19f259e2ece869e6","e-42a9be246d53e431"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-d959d759298e1e52","kind":"fingerprint_related","node_a_id":"n-439ac61e3f209ac0","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-439ac61e3f209ac0","other":{"id":"n-439ac61e3f209ac0","name":"basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["6","36","72","48"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609"],"edges":["e-456dc355a6d78eff"],"text":"A136572 \u2192 basis A013609"},"headline":"A136572 \u2192 basis A013609","chain":null}},{"id":"l-dbc9db0d9aae6f80","kind":"fingerprint_related","node_a_id":"n-5d91f73259c0b712","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5d91f73259c0b712","other":{"id":"n-5d91f73259c0b712","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["40","40","10"]],"card_rows":[["1"],["8","4"],["40","40","10"],["160","240","120","20"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318"],"edges":["e-2fa6de2302a1c958"],"text":"A178820 \u2192 basis A007318"},"headline":"A178820 \u2192 basis A007318","chain":null}},{"id":"l-df22139aa5967754","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-a4124af3cb94889c","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a4124af3cb94889c","other":{"id":"n-a4124af3cb94889c","name":"basis_map_A013609 of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["18","72","72"]],"card_rows":[["1"],["4","8"],["18","72","72"],["96","576","1152","768"]],"row_sums_preview":["1","12","162","2592","48600","1049760","25719120","705438720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-76872140ee232c50","e-8f82839c781e539f"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-dfade47bab0885aa","kind":"fingerprint_related","node_a_id":"n-0be19de8f4a58a03","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0be19de8f4a58a03","other":{"id":"n-0be19de8f4a58a03","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["16","64","64"]],"card_rows":[["1"],["4","8"],["16","64","64"],["64","384","768","512"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611"],"edges":["e-e6fcabb7b5383fc4"],"text":"A038208 \u2192 basis A013611"},"headline":"A038208 \u2192 basis A013611","chain":null}},{"id":"l-e17f50398ecb0529","kind":"fingerprint_related","node_a_id":"n-8e586443bf59374d","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8e586443bf59374d","other":{"id":"n-8e586443bf59374d","name":"reversal of shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","4"],["2","8","8"]],"card_rows":[["2"],["2","4"],["2","8","8"],["2","12","24","16"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","reverse"],"edges":["e-014dd8a0f9594ad5","e-4732cb9c59a6ea9c"],"text":"A028326 \u2192 basis A007318 \u2192 reverse"},"headline":"A028326 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-e1aaba5852cd1122","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-abd65809ea4aa997","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-abd65809ea4aa997","other":{"id":"n-abd65809ea4aa997","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["32","32","8"]],"card_rows":[["1"],["4","2"],["32","32","8"],["384","576","288","48"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013609","basis A007318"],"edges":["e-6d279c69a37931ca","e-0ce129682829b50b"],"text":"A196347 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A196347 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-e2a98705e91de14c","kind":"fingerprint_related","node_a_id":"n-c80fa23f7424bb55","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c80fa23f7424bb55","other":{"id":"n-c80fa23f7424bb55","name":"basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609"],"edges":["e-7869781294d5f3a8"],"text":"basis-map coefficients of A026729 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609","chain":null}},{"id":"l-e45e8625ef7b74d1","kind":"fingerprint_related","node_a_id":"n-ce701caf7dd0b8e7","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ce701caf7dd0b8e7","other":{"id":"n-ce701caf7dd0b8e7","name":"basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["48","288","576","384"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611"],"edges":["e-b8ab5069d7db191f"],"text":"A196347 \u2192 basis A013611"},"headline":"A196347 \u2192 basis A013611","chain":null}},{"id":"l-e52f380cd35969a3","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-3d2b66ed54d0172b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3d2b66ed54d0172b","other":{"id":"n-3d2b66ed54d0172b","name":"derivative of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["96","192"]],"card_rows":[["8"],["96","192"],["1152","4608","4608"],["15360","92160","184320","122880"]],"row_sums_preview":["0","8","288","10368","414720","18662400","940584960","52672757760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611","D"],"edges":["e-413ab29272bdfecc","e-0c4c97a3fd401844"],"text":"A156992 \u2192 basis A013611 \u2192 D"},"headline":"A156992 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-e5886eb6c2ba7f17","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2f7b77f9de6690e6","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2f7b77f9de6690e6","other":{"id":"n-2f7b77f9de6690e6","name":"basis_map_A013609 of theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["21","84","84"]],"card_rows":[["1"],["8","16"],["21","84","84"],["40","240","480","320"]],"row_sums_preview":["1","24","189","1080","5265","23328","96957","384912"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["\u03b8+1","basis A013609"],"edges":["e-5734274d793ba02f","e-dae406d3dd7bd094"],"text":"A131431 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"A131431 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-e689dc96a8b92575","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-e40b7dadb62024d8","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e40b7dadb62024d8","other":{"id":"n-e40b7dadb62024d8","name":"basis_map_A007318 of Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["80","120","60","10"]],"row_sums_preview":["1","3","27","270","2835","30618","336798","3752892"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9a3e4fcd8c6a0286","steps":["basis A007318"],"edges":["e-111ea29812f86aaf"],"text":"A258758 \u2192 basis A007318"},"headline":"A258758 \u2192 basis A007318","chain":null}},{"id":"l-e69501a978cd486a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-27a014bb088e0149","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-27a014bb088e0149","other":{"id":"n-27a014bb088e0149","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","60"],["540","2160","2160"]],"card_rows":[["1"],["30","60"],["540","2160","2160"],["7560","45360","90720","60480"]],"row_sums_preview":["1","90","4860","204120","7348320","238085568","7142567040","202032610560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","basis A013610"],"edges":["e-4faaf8fb320b6fca","e-7fb5b40379d13ba8"],"text":"A178821 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-e827668a255db880","kind":"fingerprint_related","node_a_id":"n-0b43db26e266f660","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0b43db26e266f660","other":{"id":"n-0b43db26e266f660","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["160","240","120","20"]],"row_sums_preview":["1","6","54","540","5670","61236","673596","7505784"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1"],"edges":["e-0f8327ed3206c9f7"],"text":"A356546 \u2192 x \u21a6 x+1"},"headline":"A356546 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e8fc6c6738eeda68","kind":"fingerprint_related","node_a_id":"n-076de49b14800916","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-076de49b14800916","other":{"id":"n-076de49b14800916","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["12","24"]],"card_rows":[["6"],["12","24"],["18","72","72"],["24","144","288","192"]],"row_sums_preview":["0","6","36","162","648","2430","8748","30618"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["D"],"edges":["e-13c4b00cf881ee86"],"text":"A120909 \u2192 D"},"headline":"A120909 \u2192 D","chain":null}},{"id":"l-e994e782314c4d99","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-137321a5a22acf3d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-137321a5a22acf3d","other":{"id":"n-137321a5a22acf3d","name":"derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["16"],["160","320"]],"card_rows":[["16"],["160","320"],["960","3840","3840"],["4480","26880","53760","35840"]],"row_sums_preview":["0","16","480","8640","120960","1451520","15676416","156764160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","D"],"edges":["e-159425c642d412bb","e-477175b39ea2a1db"],"text":"A178820 \u2192 basis A013611 \u2192 D"},"headline":"A178820 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-e9b13517aadad6c4","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-cb530ec7666e917b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-cb530ec7666e917b","other":{"id":"n-cb530ec7666e917b","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["96","96","24"]],"card_rows":[["1"],["8","4"],["96","96","24"],["1280","1920","960","160"]],"row_sums_preview":["1","12","216","4320","90720","1959552","43110144","960740352"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","reverse"],"edges":["e-6d76bdb8087cd7e0","e-00976d096d909df0"],"text":"A356546 \u2192 basis A013611 \u2192 reverse"},"headline":"A356546 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-ea323132119765ef","kind":"fingerprint_related","node_a_id":"n-c997246ba459ce1a","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c997246ba459ce1a","other":{"id":"n-c997246ba459ce1a","name":"shift_x_plus_one of derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","4"]],"card_rows":[["1"],["8","4"],["60","60","15"],["448","672","336","56"]],"row_sums_preview":["0","1","12","135","1512","17010","192456","2189187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D","basis A007318"],"edges":["e-592ee9fddf6f118c","e-2b8102501002cba4"],"text":"A085880 \u2192 D \u2192 basis A007318"},"headline":"A085880 \u2192 D \u2192 basis A007318","chain":null}},{"id":"l-eaf3d1f4a00d31c4","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2272e70a10c82383","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2272e70a10c82383","other":{"id":"n-2272e70a10c82383","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","256","256"],["512","3072","6144","4096"],["4096","32768","98304","131072","65536"]],"card_rows":[["64","256","256"],["512","3072","6144","4096"],["4096","32768","98304","131072","65536"],["32768","327680","1310720","2621440","2621440","1048576"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","reverse"],"edges":["e-6efc8fa5932928ce","e-bd85f56230ca79fb"],"text":"A038330 \u2192 basis A013609 \u2192 reverse"},"headline":"A038330 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-ecfe7e42ad6ddec9","kind":"fingerprint_related","node_a_id":"n-15d3f71f572ae1d7","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-15d3f71f572ae1d7","other":{"id":"n-15d3f71f572ae1d7","name":"derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["40","20"]],"card_rows":[["4"],["40","20"],["240","240","60"],["1120","1680","840","140"]],"row_sums_preview":["0","4","60","540","3780","22680","122472","612360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","D"],"edges":["e-2fa6de2302a1c958","e-54e11a8f62699849"],"text":"A178820 \u2192 basis A007318 \u2192 D"},"headline":"A178820 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-ef32e543fd1cd437","kind":"fingerprint_related","node_a_id":"n-825a4f8e7a8e6bcb","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-825a4f8e7a8e6bcb","other":{"id":"n-825a4f8e7a8e6bcb","name":"derivative of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["36","18"]],"card_rows":[["4"],["36","18"],["192","192","48"],["800","1200","600","100"]],"row_sums_preview":["0","4","54","432","2700","14580","71442","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","D"],"edges":["e-18ace392e620d7bd","e-fdf09e79ee5b8d1e"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-ef9f0e496db5d617","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-2f6592cc6a610988","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2f6592cc6a610988","other":{"id":"n-2f6592cc6a610988","name":"basis_map_A013610 of basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-6"],["9","36","36"]],"card_rows":[["1"],["-3","-6"],["9","36","36"],["-27","-162","-324","-216"]],"row_sums_preview":["1","-9","81","-729","6561","-59049","531441","-4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609","basis A013610"],"edges":["e-13dcb978bba71940","e-57780a32369a4b52"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 basis A013610"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-efc98f908187227a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-33ff2c14863e09a4","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-33ff2c14863e09a4","other":{"id":"n-33ff2c14863e09a4","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","576","576"],["1728","10368","20736","13824"],["20736","165888","497664","663552","331776"]],"card_rows":[["144","576","576"],["1728","10368","20736","13824"],["20736","165888","497664","663552","331776"],["248832","2488320","9953280","19906560","19906560","7962624"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1","reverse"],"edges":["e-240272ddc3b4d83d","e-cf46c1d9f0f544f3"],"text":"A038338 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038338 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-f0555458d95d1a07","kind":"fingerprint_related","node_a_id":"n-ea171322d23eee7b","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ea171322d23eee7b","other":{"id":"n-ea171322d23eee7b","name":"derivative of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["32","64"]],"card_rows":[["8"],["32","64"],["96","384","384"],["256","1536","3072","2048"]],"row_sums_preview":["0","8","96","864","6912","51840","373248","2612736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611","D"],"edges":["e-3deac81ab612cd2e","e-8099240f6db86db6"],"text":"A028326 \u2192 basis A013611 \u2192 D"},"headline":"A028326 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-f225e1e204606146","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-447d7ea348d137e5","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-447d7ea348d137e5","other":{"id":"n-447d7ea348d137e5","name":"basis_map_A013609 of theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","36","36"],["16","96","192","128"],["25","200","600","800","400"]],"card_rows":[["9","36","36"],["16","96","192","128"],["25","200","600","800","400"],["36","360","1440","2880","2880","1152"]],"row_sums_preview":["81","432","2025","8748","35721","139968","531441","1968300"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-c527a11d264443d6","e-49e00ea72fb01b6d"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-f510266980b1c40c","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-deaae99bd89c1be0","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-deaae99bd89c1be0","other":{"id":"n-deaae99bd89c1be0","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["144","576","576"]],"card_rows":[["1"],["12","24"],["144","576","576"],["1728","10368","20736","13824"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013609","basis A013611"],"edges":["e-86b1b19edf92ec9a","e-5fca62ab9fa846f2"],"text":"A038221 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A038221 \u2192 basis A013609 \u2192 basis A013611","chain":null}},{"id":"l-f55cf92b7182e75a","kind":"fingerprint_related","node_a_id":"n-af754f07daf65b5e","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af754f07daf65b5e","other":{"id":"n-af754f07daf65b5e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["4","4","1"]],"card_rows":[["1"],["2","1"],["4","4","1"],["8","12","6","1"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038207"],"derivation":{"root":"n-af754f07daf65b5e","steps":[],"edges":[],"text":"A038207"},"headline":"A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","chain":null}},{"id":"l-f61461cd9d66e8f4","kind":"fingerprint_related","node_a_id":"n-5958cb66790da863","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5958cb66790da863","other":{"id":"n-5958cb66790da863","name":"derivative of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["24","12"]],"card_rows":[["2"],["24","12"],["240","240","60"],["2240","3360","1680","280"]],"row_sums_preview":["0","2","36","540","7560","102060","1347192","17513496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","D"],"edges":["e-0f8327ed3206c9f7","e-61f84749b47be254"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-f61722c8a397184e","kind":"fingerprint_related","node_a_id":"n-e93f0e54dbc33ad5","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e93f0e54dbc33ad5","other":{"id":"n-e93f0e54dbc33ad5","name":"basis_map_A007318 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","15"],["180","180","45"]],"card_rows":[["3"],["30","15"],["180","180","45"],["840","1260","630","105"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A007318"],"edges":["e-5d53734a383a65e0"],"text":"A344565 \u2192 basis A007318"},"headline":"A344565 \u2192 basis A007318","chain":null}},{"id":"l-f640bbc237089a31","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-bb273e5a357e194f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bb273e5a357e194f","other":{"id":"n-bb273e5a357e194f","name":"basis_map_A013611 of Numbers in (4,4)-Pascal triangle .","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["16","64","64"]],"card_rows":[["1"],["8","16"],["16","64","64"],["32","192","384","256"]],"row_sums_preview":["1","24","144","864","5184","31104","186624","1119744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3aa21f78720c1d0c","steps":["basis A013611"],"edges":["e-6adf3656deeefc84"],"text":"A132200 \u2192 basis A013611"},"headline":"A132200 \u2192 basis A013611","chain":null}},{"id":"l-f65bebbd7278d127","kind":"fingerprint_related","node_a_id":"n-71965e79e47a4526","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-71965e79e47a4526","other":{"id":"n-71965e79e47a4526","name":"Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013609"],"derivation":{"root":"n-71965e79e47a4526","steps":[],"edges":[],"text":"A013609"},"headline":"A013609: Triangle of coefficients in expansion of (1+2*x)^n.","chain":null}},{"id":"l-f70055e6306f3b2a","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-170caf7aaf900871","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-170caf7aaf900871","other":{"id":"n-170caf7aaf900871","name":"basis_map_A013609 of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["7","28","28"]],"card_rows":[["1"],["4","8"],["7","28","28"],["10","60","120","80"]],"row_sums_preview":["1","12","63","270","1053","3888","13851","48114"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["basis A013609"],"edges":["e-a3c37a89023e80b3"],"text":"A131431 \u2192 basis A013609"},"headline":"A131431 \u2192 basis A013609","chain":null}},{"id":"l-f812e08e0fccd5f4","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-f63e3afd2851ae64","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f63e3afd2851ae64","other":{"id":"n-f63e3afd2851ae64","name":"basis_map_A013609 of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["3","12","12"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["1","6","27","108","405","1458","5103","17496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["basis A013609"],"edges":["e-c611ab56b95a6885"],"text":"image under \u03b8+1 \u2192 basis A013609"},"headline":"image under \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-f8cd48f1dc4cbdb8","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-62ed3f68386e1711","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-62ed3f68386e1711","other":{"id":"n-62ed3f68386e1711","name":"basis_map_A013609 of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","4"]],"card_rows":[["-1"],["2","4"],["-3","-12","-12"],["4","24","48","32"]],"row_sums_preview":["0","-1","6","-27","108","-405","1458","-5103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","basis A013609"],"edges":["e-c22d2d4fa11fd8cc","e-11ac62a480f17c88"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-fa7971f1373a8402","kind":"fingerprint_related","node_a_id":"n-90af7b2f8947318a","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-90af7b2f8947318a","other":{"id":"n-90af7b2f8947318a","name":"mobius_x_over_x_plus_one of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","1"],["0","0","0","1","2"],["0","0","0","0","0","1","4","4"]],"card_rows":[["0","1"],["0","0","0","1","2"],["0","0","0","0","0","1","4","4"],["0","0","0","0","0","0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-07e2f0a7469aac14","steps":["x \u21a6 x/(x+1)"],"edges":["e-58e3ea1065a96094"],"text":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)"},"headline":"Veronese parent of A098172 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-fafc961b1c7d3d5b","kind":"fingerprint_related","node_a_id":"n-d3bd5be1ddc9cce2","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-d3bd5be1ddc9cce2","other":{"id":"n-d3bd5be1ddc9cce2","name":"shift_x_plus_one of derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","4"]],"card_rows":[["2"],["8","4"],["24","24","6"],["64","96","48","8"]],"row_sums_preview":["0","2","12","54","216","810","2916","10206"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","D"],"edges":["e-014dd8a0f9594ad5","e-57965c0d2c44fdd8"],"text":"A028326 \u2192 basis A007318 \u2192 D"},"headline":"A028326 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-fd59482c3d644e30","kind":"fingerprint_related","node_a_id":"n-0e583ea589690e69","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0e583ea589690e69","other":{"id":"n-0e583ea589690e69","name":"derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["60","30"]],"card_rows":[["5"],["60","30"],["420","420","105"],["2240","3360","1680","280"]],"row_sums_preview":["0","5","90","945","7560","51030","306180","1683990"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318","D"],"edges":["e-484d449f4d0f99c5","e-3de01224f31b09ea"],"text":"A178821 \u2192 basis A007318 \u2192 D"},"headline":"A178821 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-fe06c85e7c8952f3","kind":"fingerprint_related","node_a_id":"n-bfcfc0c48e00f4c8","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bfcfc0c48e00f4c8","other":{"id":"n-bfcfc0c48e00f4c8","name":"shift_x_plus_one of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["24","36","18","3"]],"row_sums_preview":["1","3","18","81","405","1944","9477","45927"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["basis A007318"],"edges":["e-22417f5eeaa55b57"],"text":"basis-map coefficients of A127647 \u2192 basis A007318"},"headline":"basis-map coefficients of A127647 \u2192 basis A007318","chain":null}},{"id":"l-fe2ccb2eee8190ea","kind":"fingerprint_related","node_a_id":"n-3287a7cc800f9f98","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3287a7cc800f9f98","other":{"id":"n-3287a7cc800f9f98","name":"mobius_x_over_x_plus_one of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","30"],["45","180","180"]],"card_rows":[["3"],["15","30"],["45","180","180"],["105","630","1260","840"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["x \u21a6 x/(x+1)"],"edges":["e-1dde439c20d60e3a"],"text":"A344565 \u2192 x \u21a6 x/(x+1)"},"headline":"A344565 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-fe96ed9fe823f6cb","kind":"fingerprint_related","node_a_id":"n-816bdb5b69d24e1f","node_b_id":"n-35bf4e217ba68979","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-35bf4e217ba68979","other":{"id":"n-35bf4e217ba68979","name":"basis_map_A013609 of theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","8"]],"card_rows":[["1"],["4","8"],["9","36","36"],["16","96","192","128"]],"row_sums_preview":["0","1","12","81","432","2025","8748","35721"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","basis A013609"],"edges":["e-0695c41826b28137","e-4053736d156f1a86"],"text":"A132440 \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"A132440 \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-fef4b58295b62c5a","kind":"fingerprint_related","node_a_id":"n-0b91678f1ee08b15","node_b_id":"n-816bdb5b69d24e1f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0b91678f1ee08b15","other":{"id":"n-0b91678f1ee08b15","name":"mobius_x_over_x_plus_one of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["7","28","28"]],"card_rows":[["1"],["4","8"],["7","28","28"],["10","60","120","80"]],"row_sums_preview":["1","12","63","270","1053","3888","13851","48114"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-28824f426538156a","e-cfc8ed0340c69f03"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:          1;\n   2:          4,          2;\n   3:         32,         32,           8;\n   4:        320,        480,         240,          40;\n   5:       3584,       7168,        5376,        1792,         224;\n   6:      43008,     107520,      107520,       53760,       13440,       1344;\n   7:     540672,    1622016,     2027520,     1351680,      506880,     101376,       8448;\n   8:    7028736,   24600576,    36900864,    30750720,    15375360,    4612608,     768768,     54912;\n   9:   93716480,  374865920,   656015360,   656015360,   410009600,  164003840,   41000960,   5857280,   366080;\n  10: 1274544128, 5735448576, 11470897152, 13382713344, 10037035008, 5018517504, 1672839168, 358465536, 44808192, 2489344;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (16*n-24+(8*n-12)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n - 24 + (8*n - 12)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n - 24 + (8*n - 12)*x)*v[i - 1])/(n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-ccc293aee69469d4","e-34077ec359de479d"],"text":"A085880 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
