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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":"86838471da2e828752b50677444c731729c8a5b155fe40aa82c89b582e30c6c4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-45d3b6c39c71e61d","role":"sequence","name":"basis_map_A008277 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:  1,     4;\n   3:  2,    16,      9;\n   4:  3,    52,     72,      16;\n   5:  5,   156,    396,     208,      25;\n   6:  8,   448,   1872,    1728,     475,      36;\n   7: 13,  1260,   8163,   11776,    5525,     936,     49;\n   8: 21,  3500,  33948,   71856,   51100,   14472,   1666,    64;\n   9: 34,  9656, 137043,  409392,  413800,  173988,  32977,  2752,   81;\n  10: 55, 26532, 542574, 2228976, 3076450, 1800216, 499016, 67776, 4293, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-4,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-4,x) - (x^2-4*x^3)*d^2/dx^2 P(n-4,x) - (8*x-18*x^2)*P(n-3,x) - (4*x-8*x^2-8*x^3)*d/dx P(n-3,x) - (2*x^2-8*x^3)*d^2/dx^2 P(n-3,x) + (1+9*x^2)*P(n-2,x) + 4*x^3*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (1+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 2 + 16*x + 9*x^2, P(4,x) = 3 + 52*x + 72*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 2 + 16*x + 9*x^2;\nP[4, x_] := 3 + 52*x + 72*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 4, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 4, x], x] - (x^2 - 4*x^3)*D[P[n - 4, x], {x, 2}] - (8*x - 18*x^2)*P[n - 3, x] - (4*x - 8*x^2 - 8*x^3)*D[P[n - 3, x], x] - (2*x^2 - 8*x^3)*D[P[n - 3, x], {x, 2}] + (1 + 9*x^2)*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (2*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] = 1 + 4*x; v[3] = 2 + 16*x + 9*x^2; v[4] = 3 + 52*x + 72*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 4] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 4], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 4], x), x) - (8*x - 18*x^2)*v[i - 3] - (4*x - 8*x^2 - 8*x^3)*deriv(v[i - 3], x) - (2*x^2 - 8*x^3)*deriv(deriv(v[i - 3], x), x) + (1 + 9*x^2)*v[i - 2] + 4*x^3*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (2*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-392f49b496f79fbf","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-08527b1121a7be71","e-48a885b6c7858e9f"],"text":"A026729 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A026729 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}