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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":"4a7a5627f4ea2695712ecf0cf605b401c261f30981fa4826298ea9066b6cd09d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,        8;\n   3:     9,       80,         96;\n   4:    27,      632,       2016,         1536;\n   5:    81,     4640,      29760,        55296,        30720;\n   6:   243,    33128,     383040,      1305600,      1689600,        737280;\n   7:   729,   233840,    4610976,     25712640,     58214400,      57507840,      20643840;\n   8:  2187,  1642712,   53526816,    459465216,   1620326400,    2719825920,    2167603200,     660602880;\n   9:  6561, 11516480,  608507520,   7748407296,  39975505920,  101443829760,  134680412160,   89841991680,   23781703680;\n  10: 19683, 80667848, 6831780480, 125962229760, 914502758400, 3292620226560, 6476798361600, 7094874931200, 4066671329280, 951268147200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+8*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + 8*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + 8*x)*v[i - 1] + (4*x + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f425461cbf5769b3","steps":["\u03b8+1"],"edges":["e-577c59a17f8036f3"],"text":"A225473 \u2192 \u03b8+1"},"headline":"A225473 \u2192 \u03b8+1","chain":null}