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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":"83d6f3c9cc6c27a47382237ffcb9eef0d161d5d493685a1a25eef6642173aa16","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-940e4d08e5526f20","role":"sequence","name":"reversal of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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: 1,  8;\n  2: 1, 16,   39;\n  3: 1, 24,  126,   160;\n  4: 1, 32,  261,   736,    605;\n  5: 1, 40,  444,  1984,   3655,   2184;\n  6: 1, 48,  675,  4160,  12270,  16416,    7651;\n  7: 1, 56,  954,  7520,  30850,  66384,   68908,   26240;\n  8: 1, 64, 1281, 12320,  65075, 195744,  327166,  275584,   88569;\n  9: 1, 72, 1656, 18816, 121905, 474264, 1111068, 1505664, 1062909, 295240;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(16*n^3+8*n^2)*x+(8*n^3-8*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(16*n^3+56*n^2+8*n-8)*x+(104*n^3+160*n^2-44*n-28)*x^2+(80*n^3+48*n^2-96*n-32)*x^3+(16*n^3-16*n)*x^4)*P(n-1,x) - ((2*n+1)*x+(8*n^2+32*n+12)*x^2+(68*n^2+140*n+36)*x^3+(64*n^2+96*n+16)*x^4+(16*n^2+16*n)*x^5)*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 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (104*n^3 + 160*n^2 - 44*n - 28)*x^2 + (80*n^3 + 48*n^2 - 96*n - 32)*x^3 + (16*n^3 - 16*n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (68*n^2 + 140*n + 36)*x^3 + (64*n^2 + 96*n + 16)*x^4 + (16*n^2 + 16*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (8*n^3 - 8*n^2)*x^2)]];\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 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (104*n^3 + 160*n^2 - 44*n - 28)*x^2 + (80*n^3 + 48*n^2 - 96*n - 32)*x^3 + (16*n^3 - 16*n)*x^4)*v[i - 1] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (68*n^2 + 140*n + 36)*x^3 + (64*n^2 + 96*n + 16)*x^4 + (16*n^2 + 16*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (8*n^3 - 8*n^2)*x^2)); 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-6466a8550c08a52f","steps":["reverse","\u03b8+1"],"edges":["e-18c37bd14c5a739f","e-cebfe5786caf751f"],"text":"A116414 \u2192 reverse \u2192 \u03b8+1"},"headline":"A116414 \u2192 reverse \u2192 \u03b8+1","chain":null}