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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":"21bb6e2e2771fa6949c76ee91934af9c979443eb919213103563711df5d5bae5","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,           4;\n  2:         10,          28,           9;\n  3:         80,         264,         135,         16;\n  4:        880,        3224,        2079,        416,         25;\n  5:      12320,       48656,       36360,       9520,       1000,        36;\n  6:     209440,      876432,      727596,     226480,      31875,      2052,      49;\n  7:    4188800,    18366400,    16523892,    5823104,     991375,     86940,    3773,     64;\n  8:   96342400,   439182400,   421373916,  163307200,   31900225,   3427200,  205114,   6400,    81;\n  9: 2504902400, 11804112000, 11943882216, 4995096384, 1084573350, 135043524, 9997764, 434304, 10206, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(12*n-16)*x-9*x^2)*P(n-2,x) + ((12*n-16)*x^2+4*x^3)*d/dx P(n-2,x) + (6*n-5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (12*n - 16)*x - 9*x^2)*P[n - 2, x] + ((12*n - 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (6*n - 5 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (12*n - 16)*x - 9*x^2)*v[i - 2] + ((12*n - 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (6*n - 5 + 4*x)*v[i - 1] - 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 - 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)\u00b2"],"edges":["e-c78003c235ea3be7"],"text":"A225470 \u2192 (\u03b8+1)\u00b2"},"headline":"A225470 \u2192 (\u03b8+1)\u00b2","chain":null}