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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":"51871039d747e5c4eda979b5f206753d9b7a9145dea61cd5304f385323164d38","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-de5746895014fee3","role":"sequence","name":"Triangle read by rows, T(n, k) = 4^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,         48;\n   4:    27,      632,       1008,         256;\n   5:    81,     4640,      14880,        9216,        1280;\n   6:   243,    33128,     191520,      217600,       70400,        6144;\n   7:   729,   233840,    2305488,     4285440,     2425600,      479232,      28672;\n   8:  2187,  1642712,   26763408,    76577536,    67513600,    22665216,    3010560,     131072;\n   9:  6561, 11516480,  304253760,  1291401216,  1665646080,   845365248,  187056128,   17825792,    589824;\n  10: 19683, 80667848, 3415890240, 20993704960, 38104281600, 27438501888, 8995553280, 1407713280, 100859904, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+8*x-16*x^2)*P(n-2,x) + (4*x-16*x^2)*d/dx P(n-2,x) + (2+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 8*x - 16*x^2)*P[n - 2, x] + (4*x - 16*x^2)*D[P[n - 2, x], x] + (2 + 8*x)*P[n - 1, x] + 4*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] = 3 + 8*x; for(n = 3, N, my(i = n); v[i] = (3 + 8*x - 16*x^2)*v[i - 2] + (4*x - 16*x^2)*deriv(v[i - 2], x) + (2 + 8*x)*v[i - 1] + 4*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-de5746895014fee3","steps":["\u03b8+1"],"edges":["e-3193c7cbf1ae6476"],"text":"A225467 \u2192 \u03b8+1"},"headline":"A225467 \u2192 \u03b8+1","chain":null}