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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":"ba914d97239f35911b522753706cf3ae86b3017c178f92342c53ca45d2a439b9","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-8b17fbdec05b5547","role":"sequence","name":"Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.","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:       6,       16;\n  3:      24,       72,       9;\n  4:     120,      384,      81;\n  5:     720,     2400,     648,     16;\n  6:    5040,    17280,    5400,    256;\n  7:   40320,   141120,   48600,   3200,    25;\n  8:  362880,  1290240,  476280,  38400,   625;\n  9: 3628800, 13063680, 5080320, 470400, 11250, 36;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*n-4)*x*P(n-3,x) + (4*n-4)*x^2*d/dx P(n-3,x) - (n^2-n-4*x)*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + 2*n*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 16*x, P(3,x) = 24 + 72*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 6 + 16*x;\nP[3, x_] := 24 + 72*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*n - 4)*x*P[n - 3, x] + (4*n - 4)*x^2*D[P[n - 3, x], x] - (n^2 - n - 4*x)*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + 2*n*P[n - 1, 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; v[3] = 6 + 16*x; v[4] = 24 + 72*x + 9*x^2; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*n - 4)*x*v[i - 3] + (4*n - 4)*x^2*deriv(v[i - 3], x) - (n^2 - n - 4*x)*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + 2*n*v[i - 1]); 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-8b17fbdec05b5547","steps":["(\u03b8+1)\u00b2"],"edges":["e-bfa4e89f4a7239cd"],"text":"A084950 \u2192 (\u03b8+1)\u00b2"},"headline":"A084950 \u2192 (\u03b8+1)\u00b2","chain":null}