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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":"48eebee499174284c3803770f71e20ddb3cd35ca442bf03ccb3f59e55fd2f42c","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,       4;\n  2:     9,      28,        9;\n  3:    27,     152,      108,       16;\n  4:    81,     768,      909,      288,       25;\n  5:   243,    3876,     6795,     3440,      625,      36;\n  6:   729,   20460,    49446,    35280,    10000,    1188,      49;\n  7:  2187,  117288,   368739,   343504,   133875,   24444,    2058,     64;\n  8:  6561,  751968,  2912922,  3349696,  1665475,  412272,   52822,   3328,   81;\n  9: 19683, 5482980, 24884334, 33781088, 20341125, 6346620, 1092798, 104064, 5103, 100;\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+(12*n-24)*x)*P(n-3,x) - (12*n-24)*x^2*d/dx P(n-3,x) - (n^2+8*n-8+(4*n+4)*x-9*x^2)*P(n-2,x) + ((4*n+4)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 28*x + 9*x^2, P(3,x) = 27 + 152*x + 108*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[2, x_] := 9 + 28*x + 9*x^2;\nP[3, x_] := 27 + 152*x + 108*x^2 + 16*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 + (12*n - 24)*x)*P[n - 3, x] - (12*n - 24)*x^2*D[P[n - 3, x], x] - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*P[n - 2, x] + ((4*n + 4)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 3 + 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] = 3 + 4*x; v[3] = 9 + 28*x + 9*x^2; v[4] = 27 + 152*x + 108*x^2 + 16*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 + (12*n - 24)*x)*v[i - 3] - (12*n - 24)*x^2*deriv(v[i - 3], x) - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*v[i - 2] + ((4*n + 4)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 3 + 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-ff69dadb0811f5e6","steps":["(\u03b8+1)\u00b2"],"edges":["e-891555cd30ebc637"],"text":"A327997 \u2192 (\u03b8+1)\u00b2"},"headline":"A327997 \u2192 (\u03b8+1)\u00b2","chain":null}