{"id":"n-8401ba73e6a78cca","revision":1,"first_index":0,"fingerprint":"f6-d47399ee7119df41","name":"basis_map_A013609 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3","2"],["15","22","8"],["105","244","188","48"],["945","3036","3644","1936","384"],["10395","42846","70444","57736","23584","3840"],["135135","681738","1429924","1595896","999424","332928","46080"],["2027025","12117960","30993192","43957376","37333904","18985856","5352192","645120"],["34459425","238437720","720771624","1243166464","1337994384","920102016","394760960","96602112","10321920"],["654729075","5150586330","17986539816","36594052048","47798889648","41565837216","24062294528","8941137408","1934954496","185794560"],["13749310575","121256894430","480729062736","1128205889328","1735657723568","1828860374496","1336624929408","669009776128","219456792576","42600775680","3715891200"],["316234143225","3091393404540","13724420120388","36524774834736","64740657207280","80248258531904","70977301615296","44792973297920","19765721304064","5807867277312","1022682562560","81749606400"],["7905853580625","84874454550900","417303944718660","1242505453757712","2495111026215664","3559982236272320","3700390745148096","2823279571215104","1569174391751680","619573993230336","164955878719488","26588121661440","1961990553600"],["213458046676875","2497162465970550","13473942325727220","44397549814143384","99673139505523440","160992407060959904","192470088262078912","172438707796658304","115772977428888064","57527032002762752","20562732549414912","5006732131565568","744264908144640","51011754393600"],["6190283353629375","78394536820098450","460664876493264780","1664799329730520296","4133652440456194512","7459627710922493536","10089419957307165760","10389884997441300352","8185700163744186368","4909927296088985600","2207076140010389504","720951743199019008","161772182020030464","22318758305464320","1428329123020800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","45","585","9945","208845","5221125","151412625","4996616625","184874815125","7579867420125","341094033905625","16713607661375625","885821206052908125","50491808745015763125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166758,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-23e30432bbaed1be"]},{"id":166759,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-23e30432bbaed1be"]},{"id":166760,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-23e30432bbaed1be"]}],"region_exact":{},"incoming_edges":[{"id":"e-23e30432bbaed1be","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-8401ba73e6a78cca","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":"ca8582feebc114ee43a528c6eeb40dce980a0ddbfdf7f768efa27cad4abf15f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-bd14a7c28004047c","kind":"oeis_match","node_a_id":"n-8401ba73e6a78cca","node_b_id":null,"fingerprint":null,"oeis_id":"A007696","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007696","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007696","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:         1;\n  1:         3,          2;\n  2:        15,         22,           8;\n  3:       105,        244,         188,          48;\n  4:       945,       3036,        3644,        1936,         384;\n  5:     10395,      42846,       70444,       57736,       23584,        3840;\n  6:    135135,     681738,     1429924,     1595896,      999424,      332928,       46080;\n  7:   2027025,   12117960,    30993192,    43957376,    37333904,    18985856,     5352192,     645120;\n  8:  34459425,  238437720,   720771624,  1243166464,  1337994384,   920102016,   394760960,   96602112,   10321920;\n  9: 654729075, 5150586330, 17986539816, 36594052048, 47798889648, 41565837216, 24062294528, 8941137408, 1934954496, 185794560;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+2*n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 1 + 2*n*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + 2*n*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7fd16a32c438d522","steps":["basis A013609"],"edges":["e-23e30432bbaed1be"],"text":"A059364 \u2192 basis A013609"},"headline":"A059364 \u2192 basis A013609","chain":null}
