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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":"b3da8d4e930068fdda1b3d9b96e9780aed28e63c0c28b2686fce3bb6064e97bd","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-2ee89feef605b3fb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,9,1) array.","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,       2;\n  2:     13,      26,       3;\n  3:     44,     132,      72,       4;\n  4:    205,     820,     720,     140,       5;\n  5:    806,    4030,    5100,    2140,     230,       6;\n  6:   3457,   20742,   33795,   20980,    4755,     342,      7;\n  7:  14168,   99176,  199008,  167480,   61880,    8928,    476,     8;\n  8:  59449,  475592, 1130952, 1194664,  611930,  146544,  15022,   632,   9;\n  9: 246410, 2217690, 6105240, 7792680, 5158260, 1776060, 299040, 23400, 810, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(160*n-160+(120*n^2+228*n-372)*x+(20*n^3+154*n^2+34*n-268)*x^2+(19*n^3+34*n^2-41*n-60)*x^3-(2*n^2+6*n+4)*x^4)*P(n-1,x) + (160+(160*n+560)*x+(40*n^2+440*n+724)*x^2+(80*n^2+404*n+410)*x^3+(41*n^2+127*n+88)*x^4+(n^2+3*n+2)*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[(-(160*n - 160 + (120*n^2 + 228*n - 372)*x + (20*n^3 + 154*n^2 + 34*n - 268)*x^2 + (19*n^3 + 34*n^2 - 41*n - 60)*x^3 - (2*n^2 + 6*n + 4)*x^4)*P[n - 1, x] + (160 + (160*n + 560)*x + (40*n^2 + 440*n + 724)*x^2 + (80*n^2 + 404*n + 410)*x^3 + (41*n^2 + 127*n + 88)*x^4 + (n^2 + 3*n + 2)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];\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] = (-(160*n - 160 + (120*n^2 + 228*n - 372)*x + (20*n^3 + 154*n^2 + 34*n - 268)*x^2 + (19*n^3 + 34*n^2 - 41*n - 60)*x^3 - (2*n^2 + 6*n + 4)*x^4)*v[i - 1] + (160 + (160*n + 560)*x + (40*n^2 + 440*n + 724)*x^2 + (80*n^2 + 404*n + 410)*x^3 + (41*n^2 + 127*n + 88)*x^4 + (n^2 + 3*n + 2)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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-7ac21ea1b3b7e282","steps":["basis A007318","\u03b8+1"],"edges":["e-a3fef8dca23d39bb","e-9c0c50daa4ab6196"],"text":"A143685 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A143685 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}