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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":"7126a64f61ee959a8aec85731fa95019a0937a7c9c9c0705e6b0231fc89f6ce6","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-f22372ef3b33093f","role":"sequence","name":"basis_map_A013610 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:      5,       6;\n  2:     22,      60,       27;\n  3:     95,     414,      405,      108;\n  4:    409,    2460,     3807,     2160,       405;\n  5:   1760,   13512,    28890,    25704,     10125,     1458;\n  6:   7573,   70740,   193833,   237600,    145800,    43740,     5103;\n  7:  32585,  358602,  1200825,  1886220,   1589625,   739206,   178605,    17496;\n  8: 140206, 1776300,  7036335, 13521600,  14584050,  9287460,  3464937,   699840,   59049;\n  9: 603275, 8646930, 39572550, 90094680, 118857375, 96722262, 49294980, 15326496, 2657205, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(156*n+90*n*x-54*n^3*x^2)*P(n,x) = -(390*n+390+(117*n^2+693*n+576)*x-(135*n^3+135*n^2-270*n-270)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (169+585*x-(117*n^2+117*n-567)*x^2-(270*n^2+270*n-135)*x^3-(81*n^2+81*n)*x^4)*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[(-(390*n + 390 + (117*n^2 + 693*n + 576)*x - (135*n^3 + 135*n^2 - 270*n - 270)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (169 + 585*x - (117*n^2 + 117*n - 567)*x^2 - (270*n^2 + 270*n - 135)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-156*n - 90*n*x + 54*n^3*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] = (-(390*n + 390 + (117*n^2 + 693*n + 576)*x - (135*n^3 + 135*n^2 - 270*n - 270)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (169 + 585*x - (117*n^2 + 117*n - 567)*x^2 - (270*n^2 + 270*n - 135)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-156*n - 90*n*x + 54*n^3*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":["basis A013610","\u03b8+1"],"edges":["e-5356faecb6c80906","e-f949f065eb2cf024"],"text":"A116414 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A116414 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}