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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":"4358d8f383f5b09693241f1bef148ce48355b86a041f7e51424aaf9ffd9887d2","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-9a6b0e2632270c41","role":"sequence","name":"basis_map_A008277 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,      6,       3;\n  1:   2,     20,      21,        4;\n  2:   4,     64,     114,       48,        5;\n  3:   8,    200,     552,      392,       90,        6;\n  4:  16,    616,    2508,     2696,     1030,      150,        7;\n  5:  32,   1880,   10956,    16824,     9550,     2286,      231,       8;\n  6:  64,   5704,   46644,    98728,    78330,    27462,     4515,     336,      9;\n  7: 128,  17240,  195132,   555832,   593390,   286230,    68159,    8184,    468,    10;\n  8: 256,  51976,  806388,  3039336,  4255130,  2715678,   879207,  151552,  13887,   630,  11;\n  9: 512, 156440, 3303516, 16271864, 29329950, 24115902, 10201947, 2368776, 309366, 22360, 825, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 6*x + 3*x^2, P(1,x) = 2 + 20*x + 21*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1 + 6*x + 3*x^2;\nP[1, x_] := 2 + 20*x + 21*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 3*x^2; v[2] = 2 + 20*x + 21*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 2*x + x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); 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 A008277","\u03b8+1"],"edges":["e-94a89e959b0863ff","e-9ab0ba903d98b6fc"],"text":"A159854 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A159854 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}