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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"5c8747b1b044f54094e88f49c3a0f05262b1e456c31ea539a4453a63fc4c9491","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-b17a463cc18bc314","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 4).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-b17a463cc18bc314","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 4).","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:       80,         124,           25,            1;\n  1:      320,        1072,          424,           41,            1;\n  2:     1280,        8896,         6160,         1080,           61,            1;\n  3:     5120,       72448,        82816,        23440,         2300,           85,           1;\n  4:    20480,      584704,      1066240,       457856,        69440,         4340,         113,          1;\n  5:    81920,     4698112,     13379584,      8391936,      1846656,       173600,        7504,        145,        1;\n  6:   327680,    37666816,    165253120,    147650560,     45325056,      6013056,      383712,      12144,      181,       1;\n  7:  1310720,   301662208,   2020704256,   2527662080,   1054151680,    189638400,    16756992,     772320,    18660,     221,     1;\n  8:  5242880,  2414608384,  24550113280,  42463297536,  23610695680,   5605473280,   658834176,   41471232,  1444080,   27500,   265,   1;\n  9: 20971520, 19322109952, 297015967744, 703962873856, 514677211136, 158142054400, 24052830208, 1985913600, 93458112, 2544080, 39160, 313, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 80 + 124*x + 25*x^2 + x^3.","mathematica":"P[0, x_] := 80 + 124*x + 25*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 80 + 124*x + 25*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (4 + x)*v[i - 1] + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b17a463cc18bc314","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-4d98a7f5f8c29383"],"text":"A075499 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075499 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}