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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":"a65a949c50948917fbaeb6703f8f5e01ecf4057eafff734c6129d81f607ec4f1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6321b10e19f62245","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,      12;\n   3:    10,      76,      108;\n   4:    33,     372,     1044,      864;\n   5:   109,    1628,     6816,    11232,      6480;\n   6:   360,    6696,    37260,    91872,    105840,     46656;\n   7:  1189,   26472,   183744,   603104,   1041120,    917568,    326592;\n   8:  3927,  101820,   846828,  3470496,   7982160,  10549440,   7511616,   2239488;\n   9: 12970,  383812,  3720024, 18279200,  52524080,  92536128,  98848512,  58973184,  15116544;\n  10: 42837, 1424604, 15766668, 90326496, 311324640, 685488384, 976425408, 874146816, 448457472, 100776960;\n","recurrence_sentence":"The row polynomials satisfy (156*n-156+(156*n^2+132*n-288)*x+(24*n^3+216*n^2-240*n)*x^2)*P(n,x) = -(104*n-676+(156*n^2-1108*n-2080)*x+(16*n^3+56*n^2-3848*n-1584)*x^2-(288*n^2+2880*n)*x^3)*P(n-1,x) + (169+(338*n+858)*x+(52*n^2+1664*n+1436)*x^2+(176*n^2+2696*n+792)*x^3+(144*n^2+1440*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(104*n - 676 + (156*n^2 - 1108*n - 2080)*x + (16*n^3 + 56*n^2 - 3848*n - 1584)*x^2 - (288*n^2 + 2880*n)*x^3)*P[n - 1, x] + (169 + (338*n + 858)*x + (52*n^2 + 1664*n + 1436)*x^2 + (176*n^2 + 2696*n + 792)*x^3 + (144*n^2 + 1440*n)*x^4)*D[P[n - 1, x], x])/(156*n - 156 + (156*n^2 + 132*n - 288)*x + (24*n^3 + 216*n^2 - 240*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(104*n - 676 + (156*n^2 - 1108*n - 2080)*x + (16*n^3 + 56*n^2 - 3848*n - 1584)*x^2 - (288*n^2 + 2880*n)*x^3)*v[i - 1] + (169 + (338*n + 858)*x + (52*n^2 + 1664*n + 1436)*x^2 + (176*n^2 + 2696*n + 792)*x^3 + (144*n^2 + 1440*n)*x^4)*deriv(v[i - 1], x))/(156*n - 156 + (156*n^2 + 132*n - 288)*x + (24*n^3 + 216*n^2 - 240*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A013609","\u03b8+1"],"edges":["e-925e2f91c3c9b55a","e-3a77089e4df4ceaa"],"text":"A099097 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A099097 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}