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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":"c144881a2a8a38519c0b89d2a5bfb15d26000d044fee69fe3d591ef94eda66c7","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-1dda68ade0f78df3","role":"sequence","name":"Triangle read by rows of Legendre-Stirling numbers of the second kind.","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,   4;\n  2: 1,  16,     12;\n  3: 1,  40,    156,      32;\n  4: 1,  80,    876,    1280,        80;\n  5: 1, 140,   3276,   15296,      9680,       192;\n  6: 1, 224,   9576,  102656,    239120,     69888,        448;\n  7: 1, 336,  23688,  485696,   2805520,   3513216,     489664,      1024;\n  8: 1, 480,  51912, 1812224,  21019120,  70845696,   49674688,   3358720,     2304;\n  9: 1, 660, 103752, 5688320, 116160880, 827534016, 1702740928, 684611584, 22673664, 5120;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+2+(2*n^3+2*n^2-4*n)*x+(n^5+2*n^4-n^3-2*n^2)*x^2)*P(n-2,x) + ((2*n^2+n-6)*x^2+(2*n^4+4*n^3-2*n^2-4*n)*x^3)*d/dx P(n-2,x) - ((n+2)*x^3+(n^3+3*n^2+2*n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+2+(2*n^3+2*n^2)*x)*P(n-1,x) - (2*n^2-n)*x^2*d/dx P(n-1,x) + n*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (2*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 - n^3 - 2*n^2)*x^2)*P[n - 2, x] + ((2*n^2 + n - 6)*x^2 + (2*n^4 + 4*n^3 - 2*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^3 + (n^3 + 3*n^2 + 2*n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + 2 + (2*n^3 + 2*n^2)*x)*P[n - 1, x] - (2*n^2 - n)*x^2*D[P[n - 1, x], x] + n*x^3*D[P[n - 1, x], {x, 2}])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 2 + (2*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 - n^3 - 2*n^2)*x^2)*v[i - 2] + ((2*n^2 + n - 6)*x^2 + (2*n^4 + 4*n^3 - 2*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^3 + (n^3 + 3*n^2 + 2*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + 2 + (2*n^3 + 2*n^2)*x)*v[i - 1] - (2*n^2 - n)*x^2*deriv(v[i - 1], x) + n*x^3*deriv(deriv(v[i - 1], x), x))/(n)); 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-1dda68ade0f78df3","steps":["\u03b8+1"],"edges":["e-636bbae1f6fcd6d4"],"text":"A191935 \u2192 \u03b8+1"},"headline":"A191935 \u2192 \u03b8+1","chain":null}