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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"06184e78af51eed3b30264a9fa5acb3a635adfa909e8393174883e95043102ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8cf5a16a6dfee934","role":"sequence","name":"Triangle read by rows, k!*2^k*S_2(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":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-8cf5a16a6dfee934","role":"sequence","name":"Triangle read by rows, k!*2^k*S_2(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":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: 2,     1;\n  1: 2,     5,       2;\n  2: 2,    17,      20,        6;\n  3: 2,    53,     134,      102,        24;\n  4: 2,   161,     776,     1116,       624,       120;\n  5: 2,   485,    4202,    10140,     10080,      4440,       720;\n  6: 2,  1457,   21980,    83586,    131280,     99240,     36000,      5040;\n  7: 2,  4373,  112814,   651042,   1515864,   1748040,   1063440,    327600,     40320;\n  8: 2, 13121,  572816,  4895736,  16246944,  26807760,  24312960,  12358080,   3306240,   362880;\n  9: 2, 39365, 2890322, 35988600, 165805440, 376120080, 476915040, 355561920, 155070720, 36650880, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2*x + x^2)*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] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (2*x + x^2)*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-8cf5a16a6dfee934","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b0c8196dbc01af81"],"text":"A225476 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225476 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}