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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":"52011de1f1e0d79357db61534dda29ab3f789736c7be6e1d48a64fb3e951d5b4","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-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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:     3,       2;\n  2:     9,      14,       3;\n  3:    27,      76,      36,       4;\n  4:    81,     384,     303,      72,       5;\n  5:   243,    1938,    2265,     860,     125,       6;\n  6:   729,   10230,   16482,    8820,    2000,     198,      7;\n  7:  2187,   58644,  122913,   85876,   26775,    4074,    294,     8;\n  8:  6561,  375984,  970974,  837424,  333095,   68712,   7546,   416,   9;\n  9: 19683, 2741490, 8294778, 8445272, 4068225, 1057770, 156114, 13008, 567, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(6*n-12)*x)*P(n-3,x) - (n^2+8*n-8+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 9 + 14*x + 3*x^2, P(3,x) = 27 + 76*x + 36*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[2, x_] := 9 + 14*x + 3*x^2;\nP[3, x_] := 27 + 76*x + 36*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 9 + 14*x + 3*x^2; v[4] = 27 + 76*x + 36*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*v[i - 3] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 1]); 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-ff69dadb0811f5e6","steps":["\u03b8+1"],"edges":["e-d0bcf3ed451f6ad7"],"text":"A327997 \u2192 \u03b8+1"},"headline":"A327997 \u2192 \u03b8+1","chain":null}