{"id":"n-8b0b17ea58a3d398","revision":0,"first_index":1,"fingerprint":"f6-a8e18dd2f0709b33","name":"mobius_x_over_x_plus_one of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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"],["1","5"],["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"],["1","30","375","2500","9375","18750","15625"],["1","35","525","4375","21875","65625","109375","78125"],["1","40","700","7000","43750","175000","437500","625000","390625"],["1","45","900","10500","78750","393750","1312500","2812500","3515625","1953125"],["1","50","1125","15000","131250","787500","3281250","9375000","17578125","19531250","9765625"],["1","55","1375","20625","206250","1443750","7218750","25781250","64453125","107421875","107421875","48828125"],["1","60","1650","27500","309375","2475000","14437500","61875000","193359375","429687500","644531250","585937500","244140625"],["1","65","1950","35750","446875","4021875","26812500","134062500","502734375","1396484375","2792968750","3808593750","3173828125","1220703125"],["1","70","2275","45500","625625","6256250","46921875","268125000","1173046875","3910156250","9775390625","17773437500","22216796875","17089843750","6103515625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","216","1296","7776","46656","279936","1679616","10077696","60466176","362797056","2176782336","13060694016","78364164096"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","6","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178463,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-394e87d0ad909faf"]},{"id":178464,"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-394e87d0ad909faf"]},{"id":178465,"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-394e87d0ad909faf"]}],"region_exact":{},"incoming_edges":[{"id":"e-394e87d0ad909faf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-8b0b17ea58a3d398","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"608df8c55ce31ae568732e010d64efde262882ca78ad467dbcf067e0afe0e421","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1e5aa25413f55f87","role":"sequence","name":"basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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-1deac5fee931f754","kind":"fingerprint_related","node_a_id":"n-6323575beb222864","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-6323575beb222864","other":{"id":"n-6323575beb222864","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","2"],["100","40","4"]],"card_rows":[["1"],["10","2"],["100","40","4"],["1000","600","120","8"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609"],"edges":["e-88ca34e9877f4f47"],"text":"A038291 \u2192 basis A013609"},"headline":"A038291 \u2192 basis A013609","chain":null}},{"id":"l-278f02a0ba64c147","kind":"fingerprint_related","node_a_id":"n-804e872f93921cf0","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-804e872f93921cf0","other":{"id":"n-804e872f93921cf0","name":"basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","3"],["225","90","9"]],"card_rows":[["1"],["15","3"],["225","90","9"],["3375","2025","405","27"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610"],"edges":["e-011a72a323f5e83e"],"text":"A147716 \u2192 basis A013610"},"headline":"A147716 \u2192 basis A013610","chain":null}},{"id":"l-2fc2fe4810c86926","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-24bd3442a6d1c2f2","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-24bd3442a6d1c2f2","other":{"id":"n-24bd3442a6d1c2f2","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["40","8"]],"card_rows":[["2"],["40","8"],["600","240","24"],["8000","4800","960","64"]],"row_sums_preview":["0","2","48","864","13824","207360","2985984","41803776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","D"],"edges":["e-88ca34e9877f4f47","e-37c5277fa0460438"],"text":"A038291 \u2192 basis A013609 \u2192 D"},"headline":"A038291 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-362139d9d4dca6b0","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-7b91a7a037fb3513","other":{"id":"n-7b91a7a037fb3513","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","1"],["25","10","1"]],"card_rows":[["1"],["5","1"],["25","10","1"],["125","75","15","1"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038243"],"derivation":{"root":"n-7b91a7a037fb3513","steps":[],"edges":[],"text":"A038243"},"headline":"A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","chain":null}},{"id":"l-3b7b3ad9deaa6d1e","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-0aa4a1e9e961a941","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-0aa4a1e9e961a941","other":{"id":"n-0aa4a1e9e961a941","name":"derivative of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["10","50"],["15","150","375"],["20","300","1500","2500"]],"card_rows":[["10","50"],["15","150","375"],["20","300","1500","2500"],["25","500","3750","12500","15625"]],"row_sums_preview":["60","540","4320","32400","233280","1632960","11197440","75582720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["D"],"edges":["e-8f637c73fe2c30f0"],"text":"A013612 \u2192 D"},"headline":"A013612 \u2192 D","chain":null}},{"id":"l-4529733542782d02","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-39158ca46e0de091","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-39158ca46e0de091","other":{"id":"n-39158ca46e0de091","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","4"],["400","160","16"]],"card_rows":[["1"],["20","4"],["400","160","16"],["8000","4800","960","64"]],"row_sums_preview":["1","24","576","13824","331776","7962624","191102976","4586471424"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-9c08db68889738c4","e-176b145beb4f8297"],"text":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-486351da414913a7","kind":"fingerprint_related","node_a_id":"n-bbfc5a54ff13a785","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-bbfc5a54ff13a785","other":{"id":"n-bbfc5a54ff13a785","name":"Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"]],"card_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":[],"edges":[],"text":"A013612"},"headline":"Triangle of coefficients in expansion of (1+5x)^n.","chain":null}},{"id":"l-528d71b2ed0bf8ca","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-af03db3dc7849a56","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-af03db3dc7849a56","other":{"id":"n-af03db3dc7849a56","name":"mobius_x_over_x_plus_one of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"]],"card_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-b91bf4c52fd06745","e-a816678ba2afa32c"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-63b2ea24dd12c21e","kind":"fingerprint_related","node_a_id":"n-b83665b9a9abd39d","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b83665b9a9abd39d","other":{"id":"n-b83665b9a9abd39d","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["x \u21a6 x/(x+1)"],"edges":["e-949a223514c284da"],"text":"A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-6961cdbd34ee3bff","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-1125779e0b7ca410","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-1125779e0b7ca410","other":{"id":"n-1125779e0b7ca410","name":"reversal 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-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","90","225"],["27","405","2025","3375"],["81","1620","12150","40500","50625"]],"card_rows":[["9","90","225"],["27","405","2025","3375"],["81","1620","12150","40500","50625"],["243","6075","60750","303750","759375","759375"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","reverse"],"edges":["e-5083e5230277d126","e-86f80e580943670b"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-6ceb82718b152f8b","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-2b94985594253b13","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-2b94985594253b13","other":{"id":"n-2b94985594253b13","name":"reversal of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["reverse"],"edges":["e-bec3e7775a5cdcbb"],"text":"A038243 \u2192 reverse"},"headline":"A038243 \u2192 reverse","chain":null}},{"id":"l-70df9e1e59327ca3","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-b9fb3905119e37a6","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b9fb3905119e37a6","other":{"id":"n-b9fb3905119e37a6","name":"reversal of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","15"],["9","90","225"]],"card_rows":[["1"],["3","15"],["9","90","225"],["27","405","2025","3375"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","reverse"],"edges":["e-011a72a323f5e83e","e-0da40532f4b4c25e"],"text":"A147716 \u2192 basis A013610 \u2192 reverse"},"headline":"A147716 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-935bc2b5945a2d73","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-9cce4140bcdbe78c","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-9cce4140bcdbe78c","other":{"id":"n-9cce4140bcdbe78c","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","160","16"],["8000","4800","960","64"],["160000","128000","38400","5120","256"]],"card_rows":[["400","160","16"],["8000","4800","960","64"],["160000","128000","38400","5120","256"],["3200000","3200000","1280000","256000","25600","1024"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","basis A007318"],"edges":["e-7b68ca495e98f7ea","e-70a0512cff30529a"],"text":"A038330 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A007318 \u2192 basis A007318","chain":null}},{"id":"l-a5dfb62e3ab3fac9","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-8f7b77214e515821","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8f7b77214e515821","other":{"id":"n-8f7b77214e515821","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["90","18"]],"card_rows":[["3"],["90","18"],["2025","810","81"],["40500","24300","4860","324"]],"row_sums_preview":["0","3","108","2916","69984","1574640","34012224","714256704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","D"],"edges":["e-011a72a323f5e83e","e-3ebd0fbcb1cb9e8c"],"text":"A147716 \u2192 basis A013610 \u2192 D"},"headline":"A147716 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-baea3fd3d01ad907","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-69e9b0dfdfe0424c","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-69e9b0dfdfe0424c","other":{"id":"n-69e9b0dfdfe0424c","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","10"],["4","40","100"]],"card_rows":[["1"],["2","10"],["4","40","100"],["8","120","600","1000"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","reverse"],"edges":["e-88ca34e9877f4f47","e-6f038ce27cd3a7a7"],"text":"A038291 \u2192 basis A013609 \u2192 reverse"},"headline":"A038291 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-c6b9d262ec4db8b8","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-c66fb08c4dfad91a","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-c66fb08c4dfad91a","other":{"id":"n-c66fb08c4dfad91a","name":"derivative 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-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["90","18"],["2025","810","81"],["40500","24300","4860","324"]],"card_rows":[["90","18"],["2025","810","81"],["40500","24300","4860","324"],["759375","607500","182250","24300","1215"]],"row_sums_preview":["108","2916","69984","1574640","34012224","714256704","14693280768","297538935552"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","D"],"edges":["e-5083e5230277d126","e-47539391ec5f1610"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-cc8c56364f76bced","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-119c10eed5405419","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-119c10eed5405419","other":{"id":"n-119c10eed5405419","name":"mobius_x_over_x_plus_one of basis_map_A013611 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-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","10"],["4","40","100"]],"card_rows":[["1"],["2","10"],["4","40","100"],["8","120","600","1000"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-d7d2422771947afc","e-6ecc5a1d7e3e25d3"],"text":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ecd520db1299dd48","kind":"fingerprint_related","node_a_id":"n-8447c9ce68adee7a","node_b_id":"n-8b0b17ea58a3d398","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8447c9ce68adee7a","other":{"id":"n-8447c9ce68adee7a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"]],"card_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"],["759375","759375","303750","60750","6075","243"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1"],"edges":["e-5083e5230277d126"],"text":"A038329 \u2192 x \u21a6 x+1"},"headline":"A038329 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ecfb3eda7b3ea360","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-e3cadb5e194cb15c","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-e3cadb5e194cb15c","other":{"id":"n-e3cadb5e194cb15c","name":"shift_x_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["10","2"]],"card_rows":[["1"],["10","2"],["75","30","3"],["500","300","60","4"]],"row_sums_preview":["0","1","12","108","864","6480","46656","326592"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["D"],"edges":["e-37cb40ea1f980993"],"text":"A038243 \u2192 D"},"headline":"A038243 \u2192 D","chain":null}},{"id":"l-f271e795fc5d5bc4","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-4b901f65c39b0b35","other":{"id":"n-4b901f65c39b0b35","name":"reversal of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["25","10","1"],["125","75","15","1"],["625","500","150","20","1"]],"card_rows":[["25","10","1"],["125","75","15","1"],["625","500","150","20","1"],["3125","3125","1250","250","25","1"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["reverse"],"edges":["e-3920ec1a803b452d"],"text":"A013612 \u2192 reverse"},"headline":"A013612 \u2192 reverse","chain":null}},{"id":"l-f9b40bfcfbce581d","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-b758fb00deafc303","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b758fb00deafc303","other":{"id":"n-b758fb00deafc303","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["2","20","50"]],"card_rows":[["1"],["1","5"],["2","20","50"],["6","90","450","750"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-3656191e13afb7f3","e-6ec3bc6d71fb3107"],"text":"A136572 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A136572 \u2192 basis A013611 \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: 1,  5;\n   3: 1, 10,  25;\n   4: 1, 15,  75,   125;\n   5: 1, 20, 150,   500,   625;\n   6: 1, 25, 250,  1250,  3125,   3125;\n   7: 1, 30, 375,  2500,  9375,  18750,   15625;\n   8: 1, 35, 525,  4375, 21875,  65625,  109375,   78125;\n   9: 1, 40, 700,  7000, 43750, 175000,  437500,  625000,  390625;\n  10: 1, 45, 900, 10500, 78750, 393750, 1312500, 2812500, 3515625, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (1+6*x+5*x^2)*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[((1 + 6*x + 5*x^2)*P[n - 1, x])/(1 + x)]];\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] = ((1 + 6*x + 5*x^2)*v[i - 1])/(1 + x)); 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-1bf99ad40f056d33","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-f93ad378f9476e41","e-394e87d0ad909faf"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}