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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":"f7ed7b8539804f3ab9dc03d7fcb10eb6edb1ab18650f8c6f8ab0f97d1371b5ba","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a5662819bbc56ccb","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:   2;\n  2:   4,    4;\n  3:   8,   16;\n  4:  16,   48,   12;\n  5:  32,  128,   72;\n  6:  64,  320,  288,   32;\n  7: 128,  768,  960,  256;\n  8: 256, 1792, 2880, 1280,  80;\n  9: 512, 4096, 8064, 5120, 800;\n","recurrence_sentence":"The row polynomials satisfy (2*n+n^2*x)*P(n,x) = (4*n+(2*n^2+8*n-8)*x+(2*n^2+2*n-4)*x^2)*P(n-1,x) - (4*x+(2*n+12)*x^2+(4*n+8)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + (2*n^2 + 8*n - 8)*x + (2*n^2 + 2*n - 4)*x^2)*P[n - 1, x] - (4*x + (2*n + 12)*x^2 + (4*n + 8)*x^3)*D[P[n - 1, x], x])/(2*n + n^2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + (2*n^2 + 8*n - 8)*x + (2*n^2 + 2*n - 4)*x^2)*v[i - 1] - (4*x + (2*n + 12)*x^2 + (4*n + 8)*x^3)*deriv(v[i - 1], x))/(2*n + n^2*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-1bffaa6198344f93","steps":["\u03b3","\u03b8+1"],"edges":["e-babfdb0b6fa40ea8","e-f990ceceeec0d7d5"],"text":"A059473 \u2192 \u03b3 \u2192 \u03b8+1"},"headline":"A059473 \u2192 \u03b3 \u2192 \u03b8+1","chain":null}