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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"12db9ced4f8f0625cf25fe4bc739b38086f34ec55a969bdab90dad446a8c10c3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 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-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 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:           2,            2;\n   2:           5,           21,             4;\n   3:          32,          241,           151,              8;\n   4:         308,         3416,          4275,            895,             16;\n   5:        3920,        58218,        117694,          53425,           4851,            32;\n   6:       61880,      1162360,       3382354,        2615214,         560519,         25199,            64;\n   7:     1164800,     26633528,     103813200,      120080636,       45971324,       5326317,        128267,          128;\n   8:    25450880,    689131872,    3426311788,     5469829724,     3234798840,     701015740,      47665523,       646647,        256;\n   9:   632952320,  19877517040,  121687486672,   253846259260,   212185861588,   73175662990,    9744502570,    410684953,    3245395,      512;\n  10: 17649632000, 632494795520, 4642221870720, 12165089548656, 13538734563828, 6742057862196, 1470476644350, 127168539450, 3452299839, 16254367, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-8+(3*n-7)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (3*n-3+(3*n-1)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 5 + 21*x + 4*x^2.","mathematica":"P[1, x_] := 2 + 2*x;\nP[2, x_] := 5 + 21*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(3*n - 8 + (3*n - 7)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (3*n - 3 + (3*n - 1)*x)*P[n - 1, x] + (3*x - 3*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] = 2 + 2*x; v[2] = 5 + 21*x + 4*x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 8 + (3*n - 7)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (3*n - 3 + (3*n - 1)*x)*v[i - 1] + (3*x - 3*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-d0e888e78d074358","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-35685a58b6434e5d"],"text":"A290595 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A290595 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}