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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"7ee0bee76e776a0760ac7d89bd94ef67a4ff8bcef320915ab446aa98f6c66476","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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:         392,           364,             43,               1;\n  1:        2744,          5488,           1267,              71,              1;\n  2:       19208,         79576,          32095,            3255,            106,             1;\n  3:      134456,       1133272,         753571,          123235,           6965,           148,            1;\n  4:      941192,      16000264,       16958263,         4204151,         367010,         13181,          197,           1;\n  5:     6588344,     224944888,      372123787,       134674491,       17049501,        920612,        22834,         253,         1;\n  6:    46118408,    3155816776,     8039544415,      4143009535,      731407026,      55715205,      2039478,       37002,       316,        1;\n  7:   322828856,   44227553272,   171986249491,    124043811395,    29742255445,    3071445636,    155649627,     4111590,     56910,      386,      1;\n  8:  2259801992,  619508574664,  3655938792583,   3645212968551,  1165022751970,  158742972157,  10698277359,   385898667,   7696920,    83930,    463,   1;\n  9: 15818613944, 8675379847288, 77394223218907, 105721901912011, 44421009287501, 7832227582564, 682958562748, 32308602711, 870804627, 13572020, 119581, 547, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 392 + 364*x + 43*x^2 + x^3.","mathematica":"P[0, x_] := 392 + 364*x + 43*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + 7*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] = 392 + 364*x + 43*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + 7*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-fc18ad0ea0731402","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a22896546ec845f5"],"text":"A075502 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075502 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}