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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":"56e1e059f7474846a2cf76c05e7e9ee58ef3c7f20170dd02cb5b291b9194f8f4","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-7d781fe2b71cdf22","role":"sequence","name":"Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].","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:        3,        12,          3;\n  3:       15,        90,         45,         4;\n  4:      105,       840,        630,       112,         5;\n  5:      945,      9450,       9450,      2520,       225,        6;\n  6:    10395,    124740,     155925,     55440,      7425,      396,       7;\n  7:   135135,   1891890,    2837835,   1261260,    225225,    18018,     637,     8;\n  8:  2027025,  32432400,   56756700,  30270240,   6756750,   720720,   38220,   960,    9;\n  9: 34459425, 620269650, 1240539300, 771891120, 206756550, 27567540, 1949220, 73440, 1377, 10;\n","recurrence_sentence":"The row polynomials satisfy -(3-n*x)*P(n,x) = -(6*n-3-(2*n^2-n-6)*x-(n+1)*x^2)*P(n-1,x) - (6*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*P[n - 1, x] - (6*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(-3 + n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*v[i - 1] - (6*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(-3 + n*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-7d781fe2b71cdf22","steps":["\u03b8+1"],"edges":["e-4fe1413713ab4b51"],"text":"A176230 \u2192 \u03b8+1"},"headline":"A176230 \u2192 \u03b8+1","chain":null}