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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":"a8c6fb01fdfe208a2d2c23cbc98de3f6dd6b82195c93d34ee006ccfb66b37ffa","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-8229dbdd6a4b5cd9","role":"sequence","name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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,          2;\n  2:         10,         14,          3;\n  3:         80,        132,         45,          4;\n  4:        880,       1612,        693,        104,         5;\n  5:      12320,      24328,      12120,       2380,       200,        6;\n  6:     209440,     438216,     242532,      56620,      6375,      342,       7;\n  7:    4188800,    9183200,    5507964,    1455776,    198275,    14490,     539,     8;\n  8:   96342400,  219591200,  140457972,   40826800,   6380045,   571200,   29302,   800,    9;\n  9: 2504902400, 5902056000, 3981294072, 1248774096, 216914670, 22507254, 1428252, 54288, 1134, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(6*n-8)*x+x^2)*P(n-2,x) + (6*n-5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (6*n - 8)*x + x^2)*P[n - 2, x] + (6*n - 5 + 2*x)*P[n - 1, 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (6*n - 8)*x + x^2)*v[i - 2] + (6*n - 5 + 2*x)*v[i - 1]); 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-8229dbdd6a4b5cd9","steps":["\u03b8+1"],"edges":["e-36c0b7f2d7fefe20"],"text":"A225470 \u2192 \u03b8+1"},"headline":"A225470 \u2192 \u03b8+1","chain":null}