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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"bcc83d425b6a22a761745c88237f9f6db9e74dcde6635a4638d3b321a5daaf0d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"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:     36,       165,        150,          27;\n   2:    108,       867,       1551,         753,           81;\n   3:    324,      4197,      12816,       12132,         3552,          243;\n   4:    972,     19299,      92733,      148278,        84798,        16071,           729;\n   5:   2916,     85701,     614586,     1527393,      1471848,       549087,         70602,         2187;\n   6:   8748,    371235,    3828411,    13999719,     20857815,     13142457,       3365025,       303549,         6561;\n   7:  26244,   1578981,   22784532,   117920754,    256611936,    249753840,     108606708,     19790742,      1284180,       19683;\n   8:  78732,   6624291,  130990089,   932074692,   2844961140,   4031458914,    2713128054,    846252516,    112778328,     5366355,      59049;\n   9: 236196,  27500997,  733164318,  7013849751,  29130163344,  57619176930,   56432679156,  27349109982,   6298587324,   627010281,   22213374,   177147;\n  10: 708588, 113251683, 4017987879, 50769569613, 280236727533, 749647232382, 1022672390502, 722344716570, 259901467302, 45204146607, 3418478595, 91274505, 531441;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(n+2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (2+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 36 + 165*x + 150*x^2 + 27*x^3, P(2,x) = 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4.","mathematica":"P[1, x_] := 36 + 165*x + 150*x^2 + 27*x^3;\nP[2, x_] := 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4;\nP[n_, x_] := P[n, x] = Expand[(3 + (n + 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (2 + (n + 4)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 36 + 165*x + 150*x^2 + 27*x^3; v[2] = 108 + 867*x + 1551*x^2 + 753*x^3 + 81*x^4; for(n = 3, N, my(i = n); v[i] = (3 + (n + 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (2 + (n + 4)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 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-b58ba5723c3a3a3f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-52155003e339a993"],"text":"A257180 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257180 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}