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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"f6337a7655abb2d275e988c89bc90283782ebbd42ba0c570d9eff3e587b524d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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:       150,          190,            31,             1;\n  1:       750,         2050,           655,            51,             1;\n  2:      3750,        21250,         11875,          1675,            76,            1;\n  3:     18750,       216250,        199375,         45375,          3575,          106,           1;\n  4:     93750,      2181250,       3206875,       1106875,        134750,         6755,         141,          1;\n  5:    468750,     21906250,      50284375,      25344375,       4475625,       337400,       11690,        181,         1;\n  6:   2343750,    219531250,     776171875,     557171875,     137235000,     14597625,      746550,      18930,       226,       1;\n  7:  11718750,   2197656250,   11862109375,   11919609375,    3988046875,    575163750,    40726875,    1503750,     29100,     276,     1;\n  8:  58593750,  21988281250,  180129296875,  250254296875,  111620781250,  21242959375,  2000604375,  100876875,   2813250,   42900,   331,   1;\n  9: 292968750, 219941406250, 2723927734375, 5185215234375, 3040773828125, 748909562500, 91264112500, 6035679375, 227473125, 4958250, 61105, 391, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 150 + 190*x + 31*x^2 + x^3.","mathematica":"P[0, x_] := 150 + 190*x + 31*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + 5*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] = 150 + 190*x + 31*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + 5*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-77f76afdd288bbc1","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-206745053438cac1"],"text":"A075500 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075500 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}