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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"733f97e485daa7e7dc0062b2db6bdfdbe32ab7dd33d4873f601b55b5d247db2d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-bdd5560e7eb65b04","role":"sequence","name":"Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 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-bdd5560e7eb65b04","role":"sequence","name":"Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 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:    3,        3;\n   2:    6,       24,         9;\n   3:   12,      138,       144,         27;\n   4:   24,      726,      1566,        729,         81;\n   5:   48,     3702,     14706,      12717,       3321,        243;\n   6:   96,    18654,    128754,     184005,      84645,      14094,        729;\n   7:  192,    93558,   1085994,    2410317,    1737045,     493533,      56862,      2187;\n   8:  384,   468366,   8968626,   29771469,   31549581,   13601196,    2617839,    220887,     6561;\n   9:  768,  2342982,  73154106,  354392037,  531008541,  325869075,   93160368,  12933918,   833247,   19683;\n  10: 1536, 11717214, 592261794, 4117774725, 8497295685, 7132799898, 2840814585, 576961218, 60466176, 3070548, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 3 + 3*x.","mathematica":"P[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x] + 3*x*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] = 3 + 3*x; for(n = 2, N, my(i = n); v[i] = (2 + 3*x)*v[i - 1] + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-bdd5560e7eb65b04","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-1a132a11ecb2f34a"],"text":"A225466 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225466 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}