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(elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f0a8cfb69f500b9f74967c3aed93fcf6fcb00e05b41d800a960be3e03bcbbbf2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e63fe40b997d2bc1","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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":[],"oeis":[],"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:             6,             12;\n  2:            66,            276,             288;\n  3:          1056,           6792,           14544,            10368;\n  4:         22176,         193320,          631440,           915840,           497664;\n  5:        576576,        6356880,        28016640,         61698240,         67889664,         29859840;\n  6:      17873856,      238576752,      1326211200,       3929843520,       6546852864,       5813710848,        2149908480;\n  7:     643458816,    10090166976,     67784050368,     252876107520,     565793558784,     759229231104,      565748416512,      180592312320;\n  8:   26381811456,   475468892352,   3747802094784,   16875189243648,   47473642232064,   85444580118528,    96081691262976,    61716132790272,   17336861982720;\n  9: 1213563326976, 24720804685440, 223749536734080, 1181021331444480, 4006307980988928, 9057604046515200, 13647772450897920, 13215764764753920, 7462837992554496, 1872381094133760;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5*n+1+12*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[(5*n + 1 + 12*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] = (5*n + 1 + 12*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-4d60c3dd8627c01d","steps":["basis A013611","basis 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