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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":"b7627b2ff2a2582c67b0f89701fb6854dee04a958b7b7e901a5caa02c828de40","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-edcd820194369f8f","role":"sequence","name":"basis_map_A013611 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,        8;\n  2:     22,       80,       48;\n  3:     95,      552,      720,       256;\n  4:    409,     3280,     6768,      5120,      1280;\n  5:   1760,    18016,    51360,     60928,     32000,      6144;\n  6:   7573,    94320,   344592,    563200,    460800,    184320,     28672;\n  7:  32585,   478136,  2134800,   4471040,   5024000,   3115008,   1003520,    131072;\n  8: 140206,  2368400, 12509040,  32051200,  46092800,  39137280,  19468288,   5242880,   589824;\n  9: 603275, 11529240, 70351200, 213557760, 375648000, 407586816, 276971520, 114819072, 26542080, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(208*n+160*n*x-128*n^3*x^2)*P(n,x) = -(520*n+520+(208*n^2+1232*n+1024)*x-(320*n^3+320*n^2-640*n-640)*x^2-(256*n^3+512*n^2+256*n)*x^3)*P(n-1,x) - (169+780*x-(208*n^2+208*n-1008)*x^2-(640*n^2+640*n-320)*x^3-(256*n^2+256*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[(-(520*n + 520 + (208*n^2 + 1232*n + 1024)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*P[n - 1, x] - (169 + 780*x - (208*n^2 + 208*n - 1008)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*D[P[n - 1, x], x])/(-208*n - 160*n*x + 128*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] = (-(520*n + 520 + (208*n^2 + 1232*n + 1024)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*v[i - 1] - (169 + 780*x - (208*n^2 + 208*n - 1008)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*deriv(v[i - 1], x))/(-208*n - 160*n*x + 128*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 A013611","\u03b8+1"],"edges":["e-137d124e39cf8ed1","e-2cf1d810eac27bdd"],"text":"A116414 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A116414 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}