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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":"bf98585128028f99aedd452a10abb8499f85fec96a6032e41694b63c2fcfa459","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-7d6d6baad7692ddc","role":"sequence","name":"Triangle read by rows, 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,        4;\n   3:     9,       40,         9;\n   4:    27,      316,       189,         16;\n   5:    81,     2320,      2790,        576,        25;\n   6:   243,    16564,     35910,      13600,      1375,        36;\n   7:   729,   116920,    432279,     267840,     47375,      2808,       49;\n   8:  2187,   821356,   5018139,    4786096,   1318625,    132804,     5145,     64;\n   9:  6561,  5758240,  57047580,   80712576,  32532150,   4953312,   319676,   8704,    81;\n  10: 19683, 40333924, 640479420, 1312106560, 744224250, 160772472, 15373260, 687360, 13851, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3-12*x+9*x^2)*P(n-2,x) - (24*x-12*x^2-4*x^3)*d/dx P(n-2,x) - (16*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + (8*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(3 - 12*x + 9*x^2)*P[n - 2, x] - (24*x - 12*x^2 - 4*x^3)*D[P[n - 2, x], x] - (16*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + (8*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; v[2] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = (3 - 12*x + 9*x^2)*v[i - 2] - (24*x - 12*x^2 - 4*x^3)*deriv(v[i - 2], x) - (16*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + (8*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-7d6d6baad7692ddc","steps":["(\u03b8+1)\u00b2"],"edges":["e-055a301494eedb19"],"text":"A225469 \u2192 (\u03b8+1)\u00b2"},"headline":"A225469 \u2192 (\u03b8+1)\u00b2","chain":null}