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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":"4a348cef35c4bb51e818ef4b689f32083d8461382916908f2a7e1a330ca89088","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-387e148a64b95986","role":"sequence","name":"Row reversed version of triangle A201637 (second-order Eulerian triangle).","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: 0,      2;\n  2: 0,      4,        3;\n  3: 0,     12,       24,        4;\n  4: 0,     48,      174,       88,        5;\n  5: 0,    240,     1332,     1312,      260,        6;\n  6: 0,   1440,    11124,    17600,     7260,      684,       7;\n  7: 0,  10080,   101952,   232560,   160600,    33660,    1680,      8;\n  8: 0,  80640,  1023408,  3141216,  3220100,  1174800,  139650,   3952,    9;\n  9: 0, 725760, 11201760, 44105184, 62200320, 34593000, 7437500, 538080, 9036, 10;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3-4*n^2+n+6+(2*n^3-6*n^2-2*n+6)*x+(n^3-n)*x^2)*P(n-2,x) - ((n^2-2*n-3)*x+(3*n+3)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n^2-3*n-3+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + n + 6 + (2*n^3 - 6*n^2 - 2*n + 6)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n^2 - 2*n - 3)*x + (3*n + 3)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 3*n - 3 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + n + 6 + (2*n^3 - 6*n^2 - 2*n + 6)*x + (n^3 - n)*x^2)*v[i - 2] - ((n^2 - 2*n - 3)*x + (3*n + 3)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 3*n - 3 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], 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-e2f0f447a502f269","steps":["reverse (shift +1)","\u03b8+1"],"edges":["e-5f52c7cc32f7742d","e-f4d05846f5945452"],"text":"A201637 \u2192 reverse (shift +1) \u2192 \u03b8+1"},"headline":"A201637 \u2192 reverse (shift +1) \u2192 \u03b8+1","chain":null}