{"id":"n-dd649f176537f8e7","revision":1,"first_index":1,"fingerprint":"f6-77a7ed3b1d5f257e","name":"shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[["1","2","1"],["1","3","3","1"],["1","4","6","4","1"],["1","5","10","10","5","1"],["1","6","15","20","15","6","1"],["1","7","21","35","35","21","7","1"],["1","8","28","56","70","56","28","8","1"],["1","9","36","84","126","126","84","36","9","1"],["1","10","45","120","210","252","210","120","45","10","1"],["1","11","55","165","330","462","462","330","165","55","11","1"],["1","12","66","220","495","792","924","792","495","220","66","12","1"],["1","13","78","286","715","1287","1716","1716","1287","715","286","78","13","1"],["1","14","91","364","1001","2002","3003","3432","3003","2002","1001","364","91","14","1"],["1","15","105","455","1365","3003","5005","6435","6435","5005","3003","1365","455","105","15","1"],["1","16","120","560","1820","4368","8008","11440","12870","11440","8008","4368","1820","560","120","16","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["4","8","16","32","64","128","256","512","1024","2048","4096","8192","16384","32768","65536"],"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","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160642,"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-cb9b2e4b7b317e85"]},{"id":160643,"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-cb9b2e4b7b317e85","e-d82776c4e84d5954"]},{"id":160644,"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-cb9b2e4b7b317e85","e-d82776c4e84d5954"]},{"id":160645,"position":3,"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-d82776c4e84d5954"]}],"region_exact":{},"incoming_edges":[{"id":"e-d82776c4e84d5954","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-dd649f176537f8e7","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":"98c24a8ac688332f0615cc216129eefbdaa1e69f9e18e97d8f1bb13e2c79ef83","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-5fe240122e98ffb3","role":"sequence","name":"basis-map coefficients of A scaled Hermite triangle.","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-cb9b2e4b7b317e85","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-dd649f176537f8e7","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":"92bd1890a969477e77fe2a83935d4c8e7babacba6449ad675d13d32371dccd6e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-5fe240122e98ffb3","role":"sequence","name":"basis-map coefficients of A scaled Hermite triangle.","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-035cde76ea9c004d","kind":"fingerprint_related","node_a_id":"n-089a9b4f824a1c20","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-089a9b4f824a1c20","other":{"id":"n-089a9b4f824a1c20","name":"Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","2"],["2","4","2"]],"card_rows":[["2"],["2","2"],["2","4","2"],["2","6","6","2"]],"row_sums_preview":["2","4","8","16","32","64","128","256"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A028326"],"derivation":{"root":"n-089a9b4f824a1c20","steps":[],"edges":[],"text":"A028326"},"headline":"A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","chain":null}},{"id":"l-041eea8f1dc28bc7","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-479ecb8291154a90","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-479ecb8291154a90","other":{"id":"n-479ecb8291154a90","name":"derivative of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","12"]],"card_rows":[["3"],["12","12"],["30","60","30"],["60","180","180","60"]],"row_sums_preview":["0","3","24","120","480","1680","5376","16128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","D"],"edges":["e-739fcf83152eb2a2","e-acad71a83b815d59"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 D"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 D","chain":null}},{"id":"l-048a3dbfbfa2e4e8","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-5079b06e19e537fc","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-5079b06e19e537fc","other":{"id":"n-5079b06e19e537fc","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","10"],["60","120","60"]],"card_rows":[["1"],["10","10"],["60","120","60"],["280","840","840","280"]],"row_sums_preview":["1","20","240","2240","17920","129024","860160","5406720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013609"],"edges":["e-7c97406b114f1661"],"text":"A178821 \u2192 basis A013609"},"headline":"A178821 \u2192 basis A013609","chain":null}},{"id":"l-11090146dab7843a","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-e47a609ea6f1d1be","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-e47a609ea6f1d1be","other":{"id":"n-e47a609ea6f1d1be","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["15","30","15"],["56","168","168","56"]],"row_sums_preview":["0","1","8","60","448","3360","25344","192192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D"],"edges":["e-592ee9fddf6f118c"],"text":"A085880 \u2192 D"},"headline":"A085880 \u2192 D","chain":null}},{"id":"l-172e714f68ef1ec6","kind":"fingerprint_related","node_a_id":"n-75a6475b3e851dc5","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-75a6475b3e851dc5","other":{"id":"n-75a6475b3e851dc5","name":"Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["6","12","6"]],"card_rows":[["1"],["2","2"],["6","12","6"],["24","72","72","24"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A156992"],"derivation":{"root":"n-75a6475b3e851dc5","steps":[],"edges":[],"text":"A156992"},"headline":"A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","chain":null}},{"id":"l-1c14671588eccf0c","kind":"fingerprint_related","node_a_id":"n-c1532953a6736702","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-c1532953a6736702","other":{"id":"n-c1532953a6736702","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","288","144"],["1728","5184","5184","1728"],["20736","82944","124416","82944","20736"]],"card_rows":[["144","288","144"],["1728","5184","5184","1728"],["20736","82944","124416","82944","20736"],["248832","1244160","2488320","2488320","1244160","248832"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":[],"edges":[],"text":"A038338"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","chain":null}},{"id":"l-1cf3c9acffde8563","kind":"fingerprint_related","node_a_id":"n-c7809537965f400d","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-c7809537965f400d","other":{"id":"n-c7809537965f400d","name":"Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["3","6","3"]],"card_rows":[["1"],["1","1"],["3","6","3"],["15","45","45","15"]],"row_sums_preview":["1","2","12","120","1680","30240","665280","17297280"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A085881"],"derivation":{"root":"n-c7809537965f400d","steps":[],"edges":[],"text":"A085881"},"headline":"A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","chain":null}},{"id":"l-23076e769c3dc952","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-bf02dae8d2c4cae7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-bf02dae8d2c4cae7","other":{"id":"n-bf02dae8d2c4cae7","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","22"],["484","968","484"]],"card_rows":[["1"],["22","22"],["484","968","484"],["10648","31944","31944","10648"]],"row_sums_preview":["1","44","1936","85184","3748096","164916224","7256313856","319277809664"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013609"],"edges":["e-290484f08292dc0a"],"text":"A038325 \u2192 basis A013609"},"headline":"A038325 \u2192 basis A013609","chain":null}},{"id":"l-26e06d04de092e2b","kind":"fingerprint_related","node_a_id":"n-a12ea22caf034cea","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a12ea22caf034cea","other":{"id":"n-a12ea22caf034cea","name":"Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1","2","1"]],"card_rows":[["1","2","1"],["3","9","9","3"],["6","24","36","24","6"],["10","50","100","100","50","10"]],"row_sums_preview":["0","0","4","24","96","320","960","2688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A237765"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)"],"edges":["e-ccfad80d3bf960c1"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2)"},"headline":"A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","chain":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2)"}},{"id":"l-29fb0cd4f975c603","kind":"fingerprint_related","node_a_id":"n-8064e72cf80883a7","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-8064e72cf80883a7","other":{"id":"n-8064e72cf80883a7","name":"Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","11"],["121","242","121"]],"card_rows":[["1"],["11","11"],["121","242","121"],["1331","3993","3993","1331"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038325"],"derivation":{"root":"n-8064e72cf80883a7","steps":[],"edges":[],"text":"A038325"},"headline":"A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","chain":null}},{"id":"l-3732a5082a7a4d57","kind":"fingerprint_related","node_a_id":"n-7717563017cb9e5f","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-7717563017cb9e5f","other":{"id":"n-7717563017cb9e5f","name":"Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","1","1"],["0","0","1","2","1"]],"card_rows":[["0","0","1"],["0","0","1","1"],["0","0","1","2","1"],["0","0","1","3","3","1"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":[],"edges":[],"text":"A159854"},"headline":"Riordan array (1-x,x/(1-x)).","chain":null}},{"id":"l-3af5a8e8cfcf71cf","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-d082f79c726ccef5","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-d082f79c726ccef5","other":{"id":"n-d082f79c726ccef5","name":"derivative of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["20","20"]],"card_rows":[["4"],["20","20"],["60","120","60"],["140","420","420","140"]],"row_sums_preview":["0","4","40","240","1120","4480","16128","53760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["D"],"edges":["e-9ebba9b1b14179b8"],"text":"A178820 \u2192 D"},"headline":"A178820 \u2192 D","chain":null}},{"id":"l-5004be482eb73b7c","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-72730e12480b3c9c","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-72730e12480b3c9c","other":{"id":"n-72730e12480b3c9c","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["40","80","40"]],"card_rows":[["1"],["8","8"],["40","80","40"],["160","480","480","160"]],"row_sums_preview":["1","16","160","1280","8960","57344","344064","1966080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013609"],"edges":["e-cccd7fcbfdf960fa"],"text":"A178820 \u2192 basis A013609"},"headline":"A178820 \u2192 basis A013609","chain":null}},{"id":"l-51f3306b7fd505c9","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-1a3a6cbc48d40fe3","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-1a3a6cbc48d40fe3","other":{"id":"n-1a3a6cbc48d40fe3","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["12","24","12"]],"card_rows":[["1"],["2","2"],["12","24","12"],["120","360","360","120"]],"row_sums_preview":["1","4","48","960","26880","967680","42577920","2214051840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013609"],"edges":["e-0f3734a094503340"],"text":"A085881 \u2192 basis A013609"},"headline":"A085881 \u2192 basis A013609","chain":null}},{"id":"l-5924027fdc14fc95","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-a5887fca81064c8a","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a5887fca81064c8a","other":{"id":"n-a5887fca81064c8a","name":"basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["24","48","24"]],"card_rows":[["1"],["4","4"],["24","48","24"],["160","480","480","160"]],"row_sums_preview":["1","8","96","1280","17920","258048","3784704","56229888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013609"],"edges":["e-2e1c22b656cc26bb"],"text":"A356546 \u2192 basis A013609"},"headline":"A356546 \u2192 basis A013609","chain":null}},{"id":"l-5c3af03149b686f2","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-e8f74cc6dbf727f1","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-e8f74cc6dbf727f1","other":{"id":"n-e8f74cc6dbf727f1","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).","first_index":1,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","16","8"]],"card_rows":[["1"],["2","2"],["8","16","8"],["40","120","120","40"]],"row_sums_preview":["1","4","32","320","3584","43008","540672","7028736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609"],"edges":["e-ccc293aee69469d4"],"text":"A085880 \u2192 basis A013609"},"headline":"A085880 \u2192 basis A013609","chain":null}},{"id":"l-5e3e7228e94d9755","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-d9455aa236ac0835","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-d9455aa236ac0835","other":{"id":"n-d9455aa236ac0835","name":"basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","16","8"]],"card_rows":[["1"],["2","2"],["8","16","8"],["48","144","144","48"]],"row_sums_preview":["1","4","32","384","6144","122880","2949120","82575360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013609"],"edges":["e-8cc9e74a0552ef08"],"text":"A196347 \u2192 basis A013609"},"headline":"A196347 \u2192 basis A013609","chain":null}},{"id":"l-5e90d9f2b4f187d0","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-e7173f1a262fe3ce","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-e7173f1a262fe3ce","other":{"id":"n-e7173f1a262fe3ce","name":"derivative of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["18","18"]],"card_rows":[["4"],["18","18"],["48","96","48"],["100","300","300","100"]],"row_sums_preview":["0","4","36","192","800","2880","9408","28672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","D"],"edges":["e-b78deffc64bc1942","e-7f6c7a1231c5d5aa"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 D"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 D","chain":null}},{"id":"l-5f0c4222ddcc55d8","kind":"fingerprint_related","node_a_id":"n-94ecad1fe442c5e0","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-94ecad1fe442c5e0","other":{"id":"n-94ecad1fe442c5e0","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["1","2","1"]],"card_rows":[["1"],["-1","-1"],["1","2","1"],["-1","-3","-3","-1"]],"row_sums_preview":["1","-2","4","-8","16","-32","64","-128"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318"],"edges":["e-4c344817bd191837"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318","chain":null}},{"id":"l-5fcd32be7093a272","kind":"fingerprint_related","node_a_id":"n-7b5887f058255aa1","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-7b5887f058255aa1","other":{"id":"n-7b5887f058255aa1","name":"Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["3","6","3"]],"card_rows":[["1"],["2","2"],["3","6","3"],["4","12","12","4"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A003506"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D"],"edges":["e-fd16b43297f2bb3e"],"text":"A007318 \u2192 D"},"headline":"A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","chain":"A007318 \u2192 D"}},{"id":"l-638242bc785e1f33","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-f5249e29b34d4267","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-f5249e29b34d4267","other":{"id":"n-f5249e29b34d4267","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["16","32","16"]],"card_rows":[["1"],["4","4"],["16","32","16"],["64","192","192","64"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A013611"],"edges":["e-b3c62654af8090af"],"text":"A027465 \u2192 basis A013611"},"headline":"A027465 \u2192 basis A013611","chain":null}},{"id":"l-638919433471cb9c","kind":"fingerprint_related","node_a_id":"n-a820c1859f68e71c","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a820c1859f68e71c","other":{"id":"n-a820c1859f68e71c","name":"Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["10","20","10"]],"card_rows":[["1"],["4","4"],["10","20","10"],["20","60","60","20"]],"row_sums_preview":["1","8","40","160","560","1792","5376","15360"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A178820"],"derivation":{"root":"n-a820c1859f68e71c","steps":[],"edges":[],"text":"A178820"},"headline":"A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","chain":null}},{"id":"l-67c26dba5b69fc9e","kind":"fingerprint_related","node_a_id":"n-a624e5aba8e1c39e","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a624e5aba8e1c39e","other":{"id":"n-a624e5aba8e1c39e","name":"Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["2","4","2"]],"card_rows":[["1"],["2","2"],["2","4","2"],["4","12","12","4"]],"row_sums_preview":["1","4","8","32","64","256","512","2048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A135838"],"derivation":{"root":"n-a624e5aba8e1c39e","steps":[],"edges":[],"text":"A135838"},"headline":"A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","chain":null}},{"id":"l-68589c341d1fb705","kind":"fingerprint_related","node_a_id":"n-8d88ef9de74b305f","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-8d88ef9de74b305f","other":{"id":"n-8d88ef9de74b305f","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["9","18","9"]],"card_rows":[["1"],["3","3"],["9","18","9"],["27","81","81","27"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038221"],"derivation":{"root":"n-8d88ef9de74b305f","steps":[],"edges":[],"text":"A038221"},"headline":"A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","chain":null}},{"id":"l-6a46ec4293f9b862","kind":"fingerprint_related","node_a_id":"n-a98fae1bf9eeb913","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a98fae1bf9eeb913","other":{"id":"n-a98fae1bf9eeb913","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","4","2"]],"card_rows":[["1"],["1","1"],["2","4","2"],["5","15","15","5"]],"row_sums_preview":["1","2","8","40","224","1344","8448","54912"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A085880"],"derivation":{"root":"n-a98fae1bf9eeb913","steps":[],"edges":[],"text":"A085880"},"headline":"A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","chain":null}},{"id":"l-6b7348354d8a3330","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-cf97dd8f766eb0a5","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-cf97dd8f766eb0a5","other":{"id":"n-cf97dd8f766eb0a5","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["24","48","24"]],"card_rows":[["1"],["4","4"],["24","48","24"],["192","576","576","192"]],"row_sums_preview":["1","8","96","1536","30720","737280","20643840","660602880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013609"],"edges":["e-b7289e9f518eca68"],"text":"A156992 \u2192 basis A013609"},"headline":"A156992 \u2192 basis A013609","chain":null}},{"id":"l-6eeb95dd5422bcef","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-f3fe039b87b32aba","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-f3fe039b87b32aba","other":{"id":"n-f3fe039b87b32aba","name":"basis_map_A013609 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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","30"],["180","360","180"]],"card_rows":[["3"],["30","30"],["180","360","180"],["840","2520","2520","840"]],"row_sums_preview":["3","60","720","6720","53760","387072","2580480","16220160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013609"],"edges":["e-47e5b69025eff315"],"text":"A344565 \u2192 basis A013609"},"headline":"A344565 \u2192 basis A013609","chain":null}},{"id":"l-73a0bf67f0dcad90","kind":"fingerprint_related","node_a_id":"n-d28ad9841fc73317","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-d28ad9841fc73317","other":{"id":"n-d28ad9841fc73317","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["4","8","4"]],"card_rows":[["1"],["2","2"],["4","8","4"],["8","24","24","8"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038208"],"derivation":{"root":"n-d28ad9841fc73317","steps":[],"edges":[],"text":"A038208"},"headline":"A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","chain":null}},{"id":"l-769dff38b86f029c","kind":"fingerprint_related","node_a_id":"n-dd4aa31f3d11d152","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-dd4aa31f3d11d152","other":{"id":"n-dd4aa31f3d11d152","name":"Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","15"],["45","90","45"]],"card_rows":[["3"],["15","15"],["45","90","45"],["105","315","315","105"]],"row_sums_preview":["3","30","180","840","3360","12096","40320","126720"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A344565"],"derivation":{"root":"n-dd4aa31f3d11d152","steps":[],"edges":[],"text":"A344565"},"headline":"A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","chain":null}},{"id":"l-76d4997409000624","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-f6b66860ba98a2c1","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-f6b66860ba98a2c1","other":{"id":"n-f6b66860ba98a2c1","name":"derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","2"]],"card_rows":[["-1"],["2","2"],["-3","-6","-3"],["4","12","12","4"]],"row_sums_preview":["0","-1","4","-12","32","-80","192","-448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D"],"edges":["e-3c99f80a26484c6d"],"text":"basis-map coefficients of A113214, shifted \u2192 D"},"headline":"basis-map coefficients of A113214, shifted \u2192 D","chain":null}},{"id":"l-792b90020b85aaca","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-9888e9a4337cf292","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-9888e9a4337cf292","other":{"id":"n-9888e9a4337cf292","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","12"],["84","168","84"]],"card_rows":[["1"],["12","12"],["84","168","84"],["448","1344","1344","448"]],"row_sums_preview":["1","24","336","3584","32256","258048","1892352","12976128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609"],"edges":["e-eabb9625052c8071"],"text":"A178822 \u2192 basis A013609"},"headline":"A178822 \u2192 basis A013609","chain":null}},{"id":"l-7bf8bf37c30b7a46","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-4330f616ae0ed2bc","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-4330f616ae0ed2bc","other":{"id":"n-4330f616ae0ed2bc","name":"derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","6"],["12","24","12"],["20","60","60","20"]],"row_sums_preview":["0","0","2","12","48","160","480","1344"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","D"],"edges":["e-e846abf2e486a264","e-d8d4324b02c6b5ad"],"text":"A007318 \u2192 D \u2192 D"},"headline":"A007318 \u2192 D \u2192 D","chain":null}},{"id":"l-83837ce48216822d","kind":"fingerprint_related","node_a_id":"n-79986ab5023c2764","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-79986ab5023c2764","other":{"id":"n-79986ab5023c2764","name":"Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","5"],["15","30","15"]],"card_rows":[["1"],["5","5"],["15","30","15"],["35","105","105","35"]],"row_sums_preview":["1","10","60","280","1120","4032","13440","42240"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A178821"],"derivation":{"root":"n-79986ab5023c2764","steps":[],"edges":[],"text":"A178821"},"headline":"A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","chain":null}},{"id":"l-90da695ff3cdbf2f","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-fe49f5f39873300f","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-fe49f5f39873300f","other":{"id":"n-fe49f5f39873300f","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","1152","576"],["13824","41472","41472","13824"],["331776","1327104","1990656","1327104","331776"]],"card_rows":[["576","1152","576"],["13824","41472","41472","13824"],["331776","1327104","1990656","1327104","331776"],["7962624","39813120","79626240","79626240","39813120","7962624"]],"row_sums_preview":["2304","110592","5308416","254803968","12230590464","587068342272","28179280429056","1352605460594688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609"],"edges":["e-9ec880266e74c89d"],"text":"A038338 \u2192 basis A013609"},"headline":"A038338 \u2192 basis A013609","chain":null}},{"id":"l-98b7578330791f3e","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-2494c903d5dafb67","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-2494c903d5dafb67","other":{"id":"n-2494c903d5dafb67","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","8"]],"card_rows":[["2"],["8","8"],["24","48","24"],["64","192","192","64"]],"row_sums_preview":["0","2","16","96","512","2560","12288","57344"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D"],"edges":["e-a49f8ff31c907f66"],"text":"A038208 \u2192 D"},"headline":"A038208 \u2192 D","chain":null}},{"id":"l-a1efda868001f98d","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-526f71cad551e1a3","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-526f71cad551e1a3","other":{"id":"n-526f71cad551e1a3","name":"derivative of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["42","42"]],"card_rows":[["6"],["42","42"],["168","336","168"],["504","1512","1512","504"]],"row_sums_preview":["0","6","84","672","4032","20160","88704","354816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["D"],"edges":["e-32f35cc681259b27"],"text":"A178822 \u2192 D"},"headline":"A178822 \u2192 D","chain":null}},{"id":"l-a60cbff860722cf8","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-cb064c72efa341c9","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-cb064c72efa341c9","other":{"id":"n-cb064c72efa341c9","name":"basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-2"],["4","8","4"]],"card_rows":[["1"],["-2","-2"],["4","8","4"],["-8","-24","-24","-8"]],"row_sums_preview":["1","-4","16","-64","256","-1024","4096","-16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013609"],"edges":["e-b6cdc8ab0f2a09ef"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013609","chain":null}},{"id":"l-aac77426306ffb18","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-e5e748a2234b1a03","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-e5e748a2234b1a03","other":{"id":"n-e5e748a2234b1a03","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["8","16","8"]],"card_rows":[["1"],["4","4"],["8","16","8"],["32","96","96","32"]],"row_sums_preview":["1","8","32","256","1024","8192","32768","262144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013609"],"edges":["e-56a3ed705651b472"],"text":"A135838 \u2192 basis A013609"},"headline":"A135838 \u2192 basis A013609","chain":null}},{"id":"l-af52752cb3f997c0","kind":"fingerprint_related","node_a_id":"n-dd77bf712b1df996","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-dd77bf712b1df996","other":{"id":"n-dd77bf712b1df996","name":"Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["6","12","6"]],"card_rows":[["1"],["2","2"],["6","12","6"],["20","60","60","20"]],"row_sums_preview":["1","4","24","160","1120","8064","59136","439296"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A356546"],"derivation":{"root":"n-dd77bf712b1df996","steps":[],"edges":[],"text":"A356546"},"headline":"A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","chain":null}},{"id":"l-b2ae4453c2f339b6","kind":"fingerprint_related","node_a_id":"n-74c1e94669ff21ff","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-74c1e94669ff21ff","other":{"id":"n-74c1e94669ff21ff","name":"basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["1","2","1"]],"card_rows":[["1"],["-1","-1"],["1","2","1"],["-1","-3","-3","-1"]],"row_sums_preview":["1","-2","4","-8","16","-32","64","-128"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":[],"edges":[],"text":"basis-map coefficients of A113214, shifted"},"headline":"basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","chain":null}},{"id":"l-b2afe0e8f7309077","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-8df915a5d54fef7f","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-8df915a5d54fef7f","other":{"id":"n-8df915a5d54fef7f","name":"derivative of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","4"]],"card_rows":[["2"],["4","4"],["12","24","12"],["16","48","48","16"]],"row_sums_preview":["0","2","8","48","128","640","1536","7168"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["D"],"edges":["e-c0bb96f00078da51"],"text":"A135838 \u2192 D"},"headline":"A135838 \u2192 D","chain":null}},{"id":"l-b30faa5e4c24f050","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-0416b5b5a5f283f7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-0416b5b5a5f283f7","other":{"id":"n-0416b5b5a5f283f7","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["36","72","36"]],"card_rows":[["1"],["6","6"],["36","72","36"],["216","648","648","216"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013609"],"edges":["e-86b1b19edf92ec9a"],"text":"A038221 \u2192 basis A013609"},"headline":"A038221 \u2192 basis A013609","chain":null}},{"id":"l-b3936d52c6630cb3","kind":"fingerprint_related","node_a_id":"n-a6f555b0e6271a68","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a6f555b0e6271a68","other":{"id":"n-a6f555b0e6271a68","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["1","2","1"]],"card_rows":[["1"],["1","1"],["1","2","1"],["1","3","3","1"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A007318"],"edges":["e-279353e75dd63f59"],"text":"basis-map coefficients of A026729 \u2192 basis A007318"},"headline":"basis-map coefficients of A026729 \u2192 basis A007318","chain":null}},{"id":"l-b6ff357e5dccf63d","kind":"fingerprint_related","node_a_id":"n-ccd988a5d83adc8b","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-ccd988a5d83adc8b","other":{"id":"n-ccd988a5d83adc8b","name":"Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["0","1"]],"card_rows":[["0","1"],["0","3","3"],["0","6","12","6"],["0","10","30","30","10"]],"row_sums_preview":["0","0","1","6","24","80","240","672"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":["A129533"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["(\u22121/2)D\u00b2+(\u22121/2+(1/2)n)D"],"edges":["e-177701ed6604099d"],"text":"A007318 \u2192 (\u22121/2)D\u00b2+(\u22121/2+(1/2)n)D"},"headline":"A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.","chain":"A007318 \u2192 (\u22121/2)D\u00b2+(\u22121/2+(1/2)n)D"}},{"id":"l-b825eea551d74b9b","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-efe593ca91bb62f5","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-efe593ca91bb62f5","other":{"id":"n-efe593ca91bb62f5","name":"shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["7","14","7"]],"card_rows":[["1"],["4","4"],["7","14","7"],["10","30","30","10"]],"row_sums_preview":["1","8","28","80","208","512","1216","2816"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-39194b37db596ae8","steps":["reverse","x \u21a6 x+1"],"edges":["e-e8da6a4618f02a9d","e-28824f426538156a"],"text":"A010054 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A010054 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-bdacb66b8b6dda0c","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-7d6e8eec34fa58e8","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-7d6e8eec34fa58e8","other":{"id":"n-7d6e8eec34fa58e8","name":"derivative of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2","2"]],"card_rows":[["2","2"],["9","18","9"],["24","72","72","24"],["50","200","300","200","50"]],"row_sums_preview":["0","0","4","36","192","800","2880","9408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","D"],"edges":["e-ccfad80d3bf960c1","e-b26125b706cf9f72"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 D"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 D","chain":null}},{"id":"l-c030d7910e57c90e","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-97ce69a237fb4e95","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-97ce69a237fb4e95","other":{"id":"n-97ce69a237fb4e95","name":"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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","9"],["27","54","27"]],"card_rows":[["3"],["9","9"],["27","54","27"],["81","243","243","81"]],"row_sums_preview":["3","18","108","648","3888","23328","139968","839808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013610"],"edges":["e-3ea97f7a94318269"],"text":"A120910 \u2192 basis A013610"},"headline":"A120910 \u2192 basis A013610","chain":null}},{"id":"l-ce833104821ebe99","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-b05662f2ce3af6a8","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-b05662f2ce3af6a8","other":{"id":"n-b05662f2ce3af6a8","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","512","256"],["4096","12288","12288","4096"],["65536","262144","393216","262144","65536"]],"card_rows":[["256","512","256"],["4096","12288","12288","4096"],["65536","262144","393216","262144","65536"],["1048576","5242880","10485760","10485760","5242880","1048576"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611"],"edges":["e-dab838a9194ad2c3"],"text":"A038330 \u2192 basis A013611"},"headline":"A038330 \u2192 basis A013611","chain":null}},{"id":"l-d1da82c4d9d0af09","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-a75ac50997f70dee","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-a75ac50997f70dee","other":{"id":"n-a75ac50997f70dee","name":"basis_map_A007318 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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["2","4","2"]],"card_rows":[["1"],["-1","-1"],["2","4","2"],["-6","-18","-18","-6"]],"row_sums_preview":["1","-2","8","-48","384","-3840","46080","-645120"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["x \u21a6 x+1"],"edges":["e-5f6cf10d49e0d311"],"text":"basis-map coefficients of A021012 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A021012 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d1e839777c0fe39a","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-fc09c8f5d575becf","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-fc09c8f5d575becf","other":{"id":"n-fc09c8f5d575becf","name":"derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","4"]],"card_rows":[["2"],["4","4"],["6","12","6"],["8","24","24","8"]],"row_sums_preview":["0","2","8","24","64","160","384","896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["D"],"edges":["e-509720341f277f4c"],"text":"A028326 \u2192 D"},"headline":"A028326 \u2192 D","chain":null}},{"id":"l-d497bdc42ad36a93","kind":"fingerprint_related","node_a_id":"n-97228a88c81561dc","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-97228a88c81561dc","other":{"id":"n-97228a88c81561dc","name":"Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["21","42","21"]],"card_rows":[["1"],["6","6"],["21","42","21"],["56","168","168","56"]],"row_sums_preview":["1","12","84","448","2016","8064","29568","101376"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A178822"],"derivation":{"root":"n-97228a88c81561dc","steps":[],"edges":[],"text":"A178822"},"headline":"A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","chain":null}},{"id":"l-d677747ed57ab5bd","kind":"fingerprint_related","node_a_id":"n-375ba152c4aebf61","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-375ba152c4aebf61","other":{"id":"n-375ba152c4aebf61","name":"A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["9","18","9"]],"card_rows":[["1"],["4","4"],["9","18","9"],["16","48","48","16"]],"row_sums_preview":["1","8","36","128","400","1152","3136","8192"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A135065"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)"],"edges":["e-b78deffc64bc1942"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2)"},"headline":"A135065: A127733 * A007318 as infinite lower triangular matrices.","chain":"A007318 \u2192 \u00d7(1+2n+n\u00b2)"}},{"id":"l-d7bc3d949e9de295","kind":"fingerprint_related","node_a_id":"n-c8c19eba9d68881f","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-c8c19eba9d68881f","other":{"id":"n-c8c19eba9d68881f","name":"binomial","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":"binomial","degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["1","2","1"]],"card_rows":[["1"],["1","1"],["1","2","1"],["1","3","3","1"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A007318"],"derivation":{"root":"n-c8c19eba9d68881f","steps":[],"edges":[],"text":"A007318"},"headline":"A007318: Binomial","chain":null}},{"id":"l-d8259d4cfcf493c0","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-652b5ddc7f7231b8","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-652b5ddc7f7231b8","other":{"id":"n-652b5ddc7f7231b8","name":"basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","4"],["8","16","8"]],"card_rows":[["2"],["4","4"],["8","16","8"],["16","48","48","16"]],"row_sums_preview":["2","8","32","128","512","2048","8192","32768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609"],"edges":["e-23f81812d1886d3d"],"text":"A028326 \u2192 basis A013609"},"headline":"A028326 \u2192 basis A013609","chain":null}},{"id":"l-dbf415b114519066","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-f8bbbe8d865f208c","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-f8bbbe8d865f208c","other":{"id":"n-f8bbbe8d865f208c","name":"derivative of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["30","30"]],"card_rows":[["5"],["30","30"],["105","210","105"],["280","840","840","280"]],"row_sums_preview":["0","5","60","420","2240","10080","40320","147840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["D"],"edges":["e-0f508833f9fd60c5"],"text":"A178821 \u2192 D"},"headline":"A178821 \u2192 D","chain":null}},{"id":"l-e5028c1158b6bd47","kind":"fingerprint_related","node_a_id":"n-7f1b09730105c8f0","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-7f1b09730105c8f0","other":{"id":"n-7f1b09730105c8f0","name":"Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["6","12","6"]],"card_rows":[["1"],["3","3"],["6","12","6"],["10","30","30","10"]],"row_sums_preview":["1","6","24","80","240","672","1792","4608"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A094305"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)"],"edges":["e-739fcf83152eb2a2"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2)"},"headline":"A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","chain":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2)"}},{"id":"l-eb38a176ed9ad399","kind":"fingerprint_related","node_a_id":"n-dd649f176537f8e7","node_b_id":"n-d52cffbb6afd7dc1","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-d52cffbb6afd7dc1","other":{"id":"n-d52cffbb6afd7dc1","name":"derivative 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":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["15"],["90","90"]],"card_rows":[["15"],["90","90"],["315","630","315"],["840","2520","2520","840"]],"row_sums_preview":["0","15","180","1260","6720","30240","120960","443520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["D"],"edges":["e-68b6a2dca4822bae"],"text":"A344565 \u2192 D"},"headline":"A344565 \u2192 D","chain":null}},{"id":"l-f6ee1456e57dd02b","kind":"fingerprint_related","node_a_id":"n-cb5d49e71e46f2ed","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-cb5d49e71e46f2ed","other":{"id":"n-cb5d49e71e46f2ed","name":"Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","4","2"]],"card_rows":[["1"],["1","1"],["2","4","2"],["6","18","18","6"]],"row_sums_preview":["1","2","8","48","384","3840","46080","645120"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A196347"],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":[],"edges":[],"text":"A196347"},"headline":"A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","chain":null}},{"id":"l-f98367a03e60df0b","kind":"fingerprint_related","node_a_id":"n-308cf60736924b4c","node_b_id":"n-dd649f176537f8e7","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-308cf60736924b4c","other":{"id":"n-308cf60736924b4c","name":"Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","2"]],"card_rows":[["1"],["2","2"],["3","6","3"],["4","12","12","4"]],"row_sums_preview":["0","1","4","12","32","80","192","448"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A216973"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D"],"edges":["e-e846abf2e486a264"],"text":"A007318 \u2192 D"},"headline":"A216973: Exponential Riordan array [x*exp(x),x].","chain":"A007318 \u2192 D"}},{"id":"l-0564b0f0165183a6","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A046055","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A046055","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2678b87bd2b75aa6","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A155559","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A155559","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2b4059d88dd94d00","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A151821","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A151821","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-40e3ab7edcd575fc","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A020707","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A020707","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8d4cd1b409724c28","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A011782","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A011782","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9d7286333cde9c3f","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A090129","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A090129","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cba31cfdbff54b23","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A131577","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A131577","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ed1da06e562c51cc","kind":"oeis_match","node_a_id":"n-dd649f176537f8e7","node_b_id":null,"fingerprint":null,"oeis_id":"A000079","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000079","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A000079","relation":"match_row_sums"},{"oeis_id":"A011782","relation":"match_row_sums"},{"oeis_id":"A020707","relation":"match_row_sums"},{"oeis_id":"A046055","relation":"match_row_sums"},{"oeis_id":"A090129","relation":"match_row_sums"},{"oeis_id":"A131577","relation":"match_row_sums"},{"oeis_id":"A151821","relation":"match_row_sums"},{"oeis_id":"A155559","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-dd649f176537f8e7","oeis_id":"A020707","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":200,"raw_size":7101}],"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,  2,  1;\n   2: 1,  3,  3,   1;\n   3: 1,  4,  6,   4,   1;\n   4: 1,  5, 10,  10,   5,   1;\n   5: 1,  6, 15,  20,  15,   6,   1;\n   6: 1,  7, 21,  35,  35,  21,   7,   1;\n   7: 1,  8, 28,  56,  70,  56,  28,   8,   1;\n   8: 1,  9, 36,  84, 126, 126,  84,  36,   9,  1;\n   9: 1, 10, 45, 120, 210, 252, 210, 120,  45, 10,  1;\n  10: 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x + x^2.","mathematica":"P[1, x_] := 1 + 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 2*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1]); 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-5fe240122e98ffb3","steps":["x \u21a6 x+1"],"edges":["e-cb9b2e4b7b317e85"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1","chain":null}