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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"0aa47b17a19925bd50c7b4524d064ea391912f742dc82a5140a6a66ef22719ad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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:       3,        6,         3;\n  2:       8,       31,        38,         15;\n  3:      30,      180,       375,        330,         105;\n  4:     144,     1198,      3644,       5215,        3570,         945;\n  5:     840,     9072,     37044,      75432,       82215,       45990,       10395;\n  6:    5760,    77292,    401624,    1079512,     1650320,     1451835,      686070,      135135;\n  7:   45360,   732960,   4670100,   15786600,    31529190,    38553900,    28423395,    11621610,     2027025;\n  8:  403200,  7663536,  58263552,  239433436,   596738120,   946655710,   964023060,   611485875,   220270050,   34459425;\n  9: 3991680, 87626880, 778495608, 3792717456, 11422233108, 22487064600, 29559309780, 25780394640, 14346607275, 4617562950, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+(n-1)*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) + (n+(n+1)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 3 + 6*x + 3*x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 3 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + (n - 1)*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] + (n + (n + 1)*x)*P[n - 1, x] + (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; v[2] = 3 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 1)*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) + (n + (n + 1)*x)*v[i - 1] + (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-a36b2e24b7a3dbb7","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-fa5fc5a793cceca0"],"text":"A112486 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A112486 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}