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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"ab57b05e4c47f2610d59347afa9d3958233399e052078ac687c7f434839b2fdc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-54aa7a9d2edaadfb","role":"sequence","name":"A triangle of numbers related to triangle A030527.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:         1;\n  1:         6,           2;\n  2:        42,          36,           3;\n  3:       336,         552,         108,           4;\n  4:      3024,        8400,        2880,         240,          5;\n  5:     30240,      132048,       70200,        9840,        450,         6;\n  6:    332640,     2173248,     1672272,      349440,      26250,       756,       7;\n  7:   3991680,    37610496,    40049856,    11664576,    1276800,     59472,    1176,      8;\n  8:  51891840,   684972288,   977562432,   380007936,   56715120,   3792096,  119952,   1728,    9;\n  9: 726485760, 13118257152, 24488956800, 12323646720, 2403324000, 215949888, 9702000, 221760, 2430, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+3*n-4+(2*n-2)*x+x^2)*P(n-2,x) - ((5*n-5)*x+5*x^2)*d/dx P(n-2,x) + (2*n+4+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 3*n - 4 + (2*n - 2)*x + x^2)*P[n - 2, x] - ((5*n - 5)*x + 5*x^2)*D[P[n - 2, x], x] + (2*n + 4 + 2*x)*P[n - 1, x] + 5*x*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] = 1; v[2] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 3*n - 4 + (2*n - 2)*x + x^2)*v[i - 2] - ((5*n - 5)*x + 5*x^2)*deriv(v[i - 2], x) + (2*n + 4 + 2*x)*v[i - 1] + 5*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-54aa7a9d2edaadfb","steps":["\u03b8+1"],"edges":["e-820ac6e1af513417"],"text":"A049374 \u2192 \u03b8+1"},"headline":"A049374 \u2192 \u03b8+1","chain":null}