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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":"f5573fb166bd243780a17257ce77ab6aa77869277c8f9fa943feb611a684779f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bdd5560e7eb65b04","role":"sequence","name":"Triangle read by rows, 3^k*S_3(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:   2,       6;\n   3:   4,      42,        27;\n   4:   8,     234,       405,        108;\n   5:  16,    1218,      4293,       2808,        405;\n   6:  32,    6186,     39825,      48060,      16200,       1458;\n   7:  64,   31122,    346437,     687960,     407025,      83106,      5103;\n   8: 128,  155994,   2911545,    8953308,    8278200,    2878092,    392931,     17496;\n   9: 256,  780738,  23994333,  110132568,  149469705,   78729084,  17931942,   1749600,   59049;\n  10: 512, 3905226, 195467985, 1307435580, 2505573000, 1876485366, 634190634, 101721744, 7440174, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+6*x-9*x^2)*P(n-2,x) + (3*x-9*x^2)*d/dx P(n-2,x) + (1+6*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 6*x - 9*x^2)*P[n - 2, x] + (3*x - 9*x^2)*D[P[n - 2, x], x] + (1 + 6*x)*P[n - 1, x] + 3*x*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; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = (2 + 6*x - 9*x^2)*v[i - 2] + (3*x - 9*x^2)*deriv(v[i - 2], x) + (1 + 6*x)*v[i - 1] + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-bdd5560e7eb65b04","steps":["\u03b8+1"],"edges":["e-5617401a7fd2c084"],"text":"A225466 \u2192 \u03b8+1"},"headline":"A225466 \u2192 \u03b8+1","chain":null}