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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":"88844ea2811d17023bc5a02d62560f1580db52244765e5dc0148838d870face8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-56eee146d6109546","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","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: 1,        2;\n  2: 1,       20,           3;\n  3: 1,      166,         123,            4;\n  4: 1,     1336,        3330,          488,            5;\n  5: 1,    10698,       77946,        32840,         1545,           6;\n  6: 1,    85596,     1717101,      1729312,       223575,        4284,          7;\n  7: 1,   684782,    36829485,     80619820,     23178935,     1216134,      10927,        8;\n  8: 1,  5478272,   780609396,   3519428680,   2010504650,   228007752,    5709760,    26384,     9;\n  9: 1, 43826194, 16458536580, 148062834896, 157078517350, 33951772092, 1823434256, 24219904, 61389, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-2,x) - ((4*n+4)*x^2-(n+1)*x^3)*d/dx P(n-2,x) - (3*n+3)*x^3*d^2/dx^2 P(n-2,x) + (1+2*n*x)*P(n-1,x) + (7*x-x^2)*d/dx P(n-1,x) + 3*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 2, x] - ((4*n + 4)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] - (3*n + 3)*x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*n*x)*P[n - 1, x] + (7*x - x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 2] - ((4*n + 4)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) - (3*n + 3)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*n*x)*v[i - 1] + (7*x - x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), 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-56eee146d6109546","steps":["\u03b8+1"],"edges":["e-ad336b55a62ac509"],"text":"A166972 \u2192 \u03b8+1"},"headline":"A166972 \u2192 \u03b8+1","chain":null}