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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"f23ab307863677a0ceef2345591160176c4ace8a12aae5ca58da11f5f20c2800","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-df97cd413ec843e8","role":"sequence","name":"Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","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-df97cd413ec843e8","role":"sequence","name":"Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","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:            5,            1;\n   2:           32,           13,            1;\n   3:          308,          172,           25,           1;\n   4:         3920,         2680,          544,          41,          1;\n   5:        61880,        49000,        12400,        1320,         61,         1;\n   6:      1164800,      1034320,       307840,       41320,       2720,        85,        1;\n   7:     25450880,     24814720,      8386840,     1337000,     111860,      5012,      113,       1;\n   8:    632952320,    667670080,    250371520,    45702720,    4572960,    261968,     8512,     145,     1;\n   9:  17649632000,  19923904000,   8154543040,  1663264960,  191709840,  13203120,   551040,   13584,   181,   1;\n  10: 545312768000, 653283030400, 288296982400, 64572334400, 8359718080, 666251040, 33566400, 1066800, 20640, 221, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-5+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3*n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 5 + x, P(2,x) = 32 + 13*x + x^2.","mathematica":"P[1, x_] := 5 + x;\nP[2, x_] := 32 + 13*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(3*n - 5 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3*n + x)*P[n - 1, x] + x*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] = 5 + x; v[2] = 32 + 13*x + x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 5 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3*n + x)*v[i - 1] + x*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-df97cd413ec843e8","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-d3f71e9704a81a95"],"text":"A035469 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A035469 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}