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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":"98bde03d1499bc15d244bc685ea6ddc7f736bec296c1d74224db6cd2dc00734a","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-eea7ec082f5ce396","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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;\n  2: 1,   4;\n  3: 1,  16;\n  4: 1,  40,    72;\n  5: 1,  80,   552;\n  6: 1, 140,  2352,   2880;\n  7: 1, 224,  7392,  33792;\n  8: 1, 336, 19152, 209408,  201600;\n  9: 1, 480, 43344, 919040, 3242880;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^5-8*n^4+23*n^3-28*n^2+12*n)*x^2*P(n-4,x) - (2*n^3-6*n^2+4*n)*x*P(n-3,x) - (n-(2*n^3-6*n^2+4*n)*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*P[n - 4, x] - (2*n^3 - 6*n^2 + 4*n)*x*P[n - 3, x] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, x])/(n - 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; v[3] = 1 + 4*x; v[4] = 1 + 16*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*v[i - 4] - (2*n^3 - 6*n^2 + 4*n)*x*v[i - 3] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 2] + (2*n - 2)*v[i - 1])/(n - 2)); 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-08ac2c6d9914754c","steps":["\u03b3","\u03b8+1"],"edges":["e-36556341d5271807","e-e2ed4f3c1aea1b11"],"text":"A154986 \u2192 \u03b3 \u2192 \u03b8+1"},"headline":"A154986 \u2192 \u03b3 \u2192 \u03b8+1","chain":null}