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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"0747301f9fa4ce73c32ad95154d69aac00f718a060e1f3dd8fbe942f7ebb399e","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-a33b47c8b957d83a","role":"sequence","name":"basis_map_A013609 of 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:          3,           4;\n  2:         18,          36,          12;\n  3:        162,         396,         216,          32;\n  4:       1944,        5400,        3780,         960,         80;\n  5:      29160,       88776,       72900,       24480,       3600,       192;\n  6:     524880,     1714608,     1578528,      635040,     126000,     12096,       448;\n  7:   11022480,    38106288,    38292912,    17545248,    4233600,    556416,     37632,    1024;\n  8:  264539520,   958640832,  1033348752,   523200384,  145469520,  23514624,   2201472,  110592,   2304;\n  9: 7142567040, 26941460544, 30776338800, 16882007040, 5235678000, 984021696, 114307200, 8017920, 311040, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(12*n-12)*x+4*x^2)*P(n-2,x) + (6*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*P[n - 2, x] + (6*n - 3 + 4*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*v[i - 2] + (6*n - 3 + 4*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":["basis A013609","\u03b8+1"],"edges":["e-0826250557dc6011","e-e06f677e20f73a5e"],"text":"A225470 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A225470 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}