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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"02e21b7dfffae9a8c10a5d8898f17bb79cd9e51dda4489ece75a5749f69a6952","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-7d6d6baad7692ddc","role":"sequence","name":"Triangle read by rows, S_4(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-7d6d6baad7692ddc","role":"sequence","name":"Triangle read by rows, S_4(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:     4,        1;\n   2:    12,       11,         1;\n   3:    36,       89,        22,          1;\n   4:   108,      659,       331,         37,         1;\n   5:   324,     4721,      4300,        886,        56,         1;\n   6:   972,    33371,     52021,      17590,      1950,        79,        1;\n   7:  2916,   234569,    605602,     315871,     54640,      3767,      106,      1;\n   8:  8748,  1644899,   6896191,    5343667,   1354031,    141281,     6629,    137,     1;\n   9: 26244, 11523041,  77503000,   87051196,  31070256,   4603494,   320264,  10876,   172,   1;\n  10: 78732, 80687531, 864056041, 1383270940, 677386060, 136950618, 13250622, 657420, 16896, 211, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + x.","mathematica":"P[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 4*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] = 4 + x; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + 4*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-7d6d6baad7692ddc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-1f2ac171cb8f6597"],"text":"A225469 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225469 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}