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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"4bb44322d192f6ba31b6d8070c654f3bbcb66d88930efc06c651c819f3563690","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8229dbdd6a4b5cd9","role":"sequence","name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle 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-8229dbdd6a4b5cd9","role":"sequence","name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle 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: 3, 1;\n 1: 12, 8, 1;\n 2: 90, 73, 16, 1;\n 3: 960, 872, 246, 27, 1;\n 4: 13200, 12970, 4271, 621, 41, 1;\n 5: 221760, 231272, 84884, 14750, 1315, 58, 1;\n 6: 4398240, 4810708, 1916832, 378099, 40930, 2472, 78, 1;\n 7: 100531200, 114387200, 48655312, 10570644, 1315664, 97615, 4263, 101, 1;\n 8: 2601244800, 3060823600, 1373917348, 322400224, 44658943, 3846409, 208222, 6886, 127, 1;\n 9: 75147072000, 91035742400, 42763968720, 10692903744, 1613681544, 155919204, 9872289, 407616, 10566, 156, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n+x)*P(n,x) = (9*n^2+6*n-3+(6*n+2)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + x.","mathematica":"P[0, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 6*n - 3 + (6*n + 2)*x + x^2)*P[n - 1, x])/(3*n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 + 6*n - 3 + (6*n + 2)*x + x^2)*v[i - 1])/(3*n + 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-8229dbdd6a4b5cd9","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-763e8ee49ac3a70c"],"text":"A225470 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225470 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}