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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":"b73f714be7e28abcc7eceb7879578807881454c361dc2a3764bb44dd55fe67fd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9401e1a9db1eb1a5","role":"sequence","name":"Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   2,        4;\n   3:   4,       32,         9;\n   4:   8,      208,       180,         16;\n   5:  16,     1280,      2628,        640,         25;\n   6:  32,     7744,     34416,      17472,       1750,        36;\n   7:  64,    46592,    430416,     410624,      79800,      4032,       49;\n   8: 128,   279808,   5269824,    8977664,    3035600,    284256,     8232,      64;\n   9: 256,  1679360,  63867456,  188921856,  105095600,  16310016,   847896,   15360,    81;\n  10: 512, 10077184, 770188032, 3891979264, 3448058400, 836358336, 69681920, 2213376, 26730, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-2,x) - (6*x-13*x^2-4*x^3)*d/dx P(n-2,x) - (6*x^2-21*x^3)*d^2/dx^2 P(n-2,x) - (x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2+4*x)*P(n-1,x) + (5*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 2, x] - (6*x - 13*x^2 - 4*x^3)*D[P[n - 2, x], x] - (6*x^2 - 21*x^3)*D[P[n - 2, x], {x, 2}] - (x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2 + 4*x)*P[n - 1, x] + (5*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 2] - (6*x - 13*x^2 - 4*x^3)*deriv(v[i - 2], x) - (6*x^2 - 21*x^3)*deriv(deriv(v[i - 2], x), x) - (x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2 + 4*x)*v[i - 1] + (5*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9401e1a9db1eb1a5","steps":["(\u03b8+1)\u00b2"],"edges":["e-f4aa70998fabd8df"],"text":"A071951 \u2192 (\u03b8+1)\u00b2"},"headline":"A071951 \u2192 (\u03b8+1)\u00b2","chain":null}