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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":"ae85b04c3b428a8c3962c051526ab2d8a32d4c96e10f42b08df840fb18d47e8d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4598f43d5daedb7a","role":"sequence","name":"reversal of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3;\n   3:     9,     2;\n   4:    27,    12;\n   5:    81,    54,     3;\n   6:   243,   216,    27;\n   7:   729,   810,   162,    4;\n   8:  2187,  2916,   810,   48;\n   9:  6561, 10206,  3645,  360,  5;\n  10: 19683, 34992, 15309, 2160, 75;\n","recurrence_sentence":"The row polynomials satisfy (27*n-27+(3*n^2-6*n+3)*x)*P(n,x) = (81*n-81+(9*n^2+18*n-63)*x+(2*n^2-2*n-4)*x^2)*P(n-1,x) - (81*x+(9*n+45)*x^2+(4*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*D[P[n - 1, x], x])/(27*n - 27 + (3*n^2 - 6*n + 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*v[i - 1] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*deriv(v[i - 1], x))/(27*n - 27 + (3*n^2 - 6*n + 3)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["reverse","\u03b8+1"],"edges":["e-035822f9cf512625","e-7c15e89eec25dd8e"],"text":"A099097 \u2192 reverse \u2192 \u03b8+1"},"headline":"A099097 \u2192 reverse \u2192 \u03b8+1","chain":null}