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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":"a90d043032d7ffdf7d4e80253c8016e3f651af0ea59b8e79b55768834a9f7d3b","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-f1ff6363886186a4","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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,           8;\n  2:         10,          56,          36;\n  3:         80,         528,         540,         128;\n  4:        880,        6448,        8316,        3328,         400;\n  5:      12320,       97312,      145440,       76160,       16000,       1152;\n  6:     209440,     1752864,     2910384,     1811840,      510000,      65664,      3136;\n  7:    4188800,    36732800,    66095568,    46584832,    15862000,    2782080,    241472,     8192;\n  8:   96342400,   878364800,  1685495664,  1306457600,   510403600,  109670400,  13127296,   819200,   20736;\n  9: 2504902400, 23608224000, 47775528864, 39960771072, 17353173600, 4321392768, 639856896, 55590912, 2612736, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(24*n-32)*x-36*x^2)*P(n-2,x) + ((24*n-32)*x^2+16*x^3)*d/dx P(n-2,x) + (6*n-5+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (24*n - 32)*x - 36*x^2)*P[n - 2, x] + ((24*n - 32)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (6*n - 5 + 8*x)*P[n - 1, x] - 8*x^2*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; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (24*n - 32)*x - 36*x^2)*v[i - 2] + ((24*n - 32)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (6*n - 5 + 8*x)*v[i - 1] - 8*x^2*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-260e022530e5976c","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-dabb8392a3e47d84","e-81cddfc173d1240c"],"text":"A286718 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A286718 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}