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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":"8e50aef65067922634a607cdf2844e1902142d59f8d223b374eb6d6d3afcfd1f","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-ca8f77e380f6014f","role":"sequence","name":"basis_map_A013610 of Riordan array (1-x,x/(1-x)).","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,    12,     27;\n  1:   2,    30,    108,     108;\n  2:   4,    72,    351,     648,     405;\n  3:   8,   168,   1026,    2700,    3240,     1458;\n  4:  16,   384,   2808,    9504,   16605,    14580,     5103;\n  5:  32,   864,   7344,   30240,   68850,    88938,    61236,    17496;\n  6:  64,  1920,  18576,   89856,  251100,   425736,   433755,   244944,    59049;\n  7: 128,  4224,  45792,  254016,  839160,  1755432,  2357586,  1977048,   944784,   196830;\n  8: 256,  9216, 110592,  691200, 2630880,  6531840, 10859184, 12037248,  8562105,  3542940,   649539;\n  9: 512, 19968, 262656, 1824768, 7853760, 22534848, 44579808, 61305984, 57749922, 35626230, 12990780, 2125764;\n","recurrence_sentence":"The row polynomials satisfy (2+(3*n+18)*x+(9*n+18)*x^2)*P(n,x) = (4+(6*n+48)*x+(27*n+117)*x^2+(27*n+81)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 27*x^2.","mathematica":"P[0, x_] := 1 + 12*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4 + (6*n + 48)*x + (27*n + 117)*x^2 + (27*n + 81)*x^3)*P[n - 1, x])/(2 + (3*n + 18)*x + (9*n + 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (6*n + 48)*x + (27*n + 117)*x^2 + (27*n + 81)*x^3)*v[i - 1])/(2 + (3*n + 18)*x + (9*n + 18)*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-7717563017cb9e5f","steps":["basis A013610","\u03b8+1"],"edges":["e-02e979b2d9fb24b7","e-b228d07f0d00390f"],"text":"A159854 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A159854 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}