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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":"e2e5617235f57996a73fab03824e7ccbb6529c52c9dfdf6ae63df7c3cdac3b44","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-1c1963609d82ab65","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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:          12,          32,           9;\n  3:         120,         368,         162,          16;\n  4:        1680,        5632,        3096,         512,         25;\n  5:       30240,      108096,       68400,       14720,       1250,        36;\n  6:      665280,     2499072,     1748016,      445440,      50500,      2592,       49;\n  7:    17297280,    67636992,    51071328,    14689024,    2009000,    140112,     4802,     64;\n  8:   518918400,  2098298880,  1684323072,   531464192,   83221600,   7096320,   334768,   8192,    81;\n  9: 17643225600, 73417835520, 61988156928, 21064134656, 3659947200, 361113984, 21040992, 715776, 13122, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(16*n-24)*x-9*x^2)*P(n-2,x) + ((16*n-24)*x^2+4*x^3)*d/dx P(n-2,x) + (8*n-8+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[-(16*n^2 - 48*n + 36 + (16*n - 24)*x - 9*x^2)*P[n - 2, x] + ((16*n - 24)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (8*n - 8 + 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] = -(16*n^2 - 48*n + 36 + (16*n - 24)*x - 9*x^2)*v[i - 2] + ((16*n - 24)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (8*n - 8 + 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-4172472abe953cea","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-2a810a02045f5dfd","e-eccd89180b92eb67"],"text":"A290319 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A290319 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}