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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"9f6ff0b37615ed6a1390a21da20ace7161a7711b27d052c4b9f116d9749d12fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-0026e6b972fa04d4","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 9).","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-0026e6b972fa04d4","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 9).","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:          810,             594,              55,                1;\n  1:         7290,           11502,            2079,               91,               1;\n  2:        65610,          214326,           67635,             5355,             136,              1;\n  3:       590490,         3923478,         2040471,           260415,           11475,            190,             1;\n  4:      5314410,        71213094,        59016195,         11415411,          776790,          21735,           253,            1;\n  5:     47829690,      1287150102,      1664650359,        469970991,        46370961,        1950480,         37674,          325,          1;\n  6:    430467210,     23216531526,     46232709795,      18583606035,      2556664236,      151696881,       4323942,        61074,        406,        1;\n  7:   3874204890,    418328034678,   1271499695991,     715242527055,    133633496655,    10748295810,     424105227,      8721270,      93960,      496,      1;\n  8:  34867844010,   7533778829094,  34748819826435,   27020230669971,   6728749876530,   714041470395,   37466925111,   1052036667,   16332030,   138600,    595,   1;\n  9: 313810596090, 135642886767702, 945751914142839, 1007477123945391, 329813975113821, 45286989277860, 3074457752388, 113213565135, 2374931097, 28806030, 197505, 703, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 810 + 594*x + 55*x^2 + x^3.","mathematica":"P[0, x_] := 810 + 594*x + 55*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(9 + x)*P[n - 1, x] + 9*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] = 810 + 594*x + 55*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (9 + x)*v[i - 1] + 9*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-0026e6b972fa04d4","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-cdccc64d5381f9ff"],"text":"A075504 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075504 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}