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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"ecda33ff2052d987c5ff75e67c7315dcb64d38a3897fc20543bdb887e51f1c0b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-28595a60eb57b37f","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of 2nd kind.","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-28595a60eb57b37f","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of 2nd kind.","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,    1;\n  2: 2,  13,   14,      1;\n  3: 2,  25,   71,     41,      1;\n  4: 2,  41,  228,    370,    122,       1;\n  5: 2,  61,  565,   1850,   1892,     365,       1;\n  6: 2,  85, 1186,   6595,  14182,    9583,    1094,       1;\n  7: 2, 113, 2219,  18851,  70497,  105315,   48281,    3281,       1;\n  8: 2, 145, 3816,  46116, 267708,  714966,  767048,  242500,    9842,     1;\n  9: 2, 181, 6153, 100500, 839664, 3543582, 7009550, 5517460, 1215782, 29525, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-2,x) - 2*x^2*d/dx P(n-2,x) + (2*n+1)*x*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 5*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 2, x] - 2*x^2*D[P[n - 2, x], x] + (2*n + 1)*x*P[n - 1, x] - 2*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; v[2] = 2 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 3)*x)*v[i - 2] - 2*x^2*deriv(v[i - 2], x) + (2*n + 1)*x*v[i - 1] - 2*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-28595a60eb57b37f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3e597de3f3fed6c2"],"text":"A039756 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A039756 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}