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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":"f775384de1164784e5cd052575f17ce7f9a94486849a3a78b286c390a4976a58","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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,  2;\n  2: 1,  4,   3;\n  3: 2, 10,  12,   4;\n  4: 1,  8,  18,  16,   5;\n  5: 2, 18,  48,  56,  30,    6;\n  6: 1, 12,  45,  80,  75,   36,   7;\n  7: 2, 26, 108, 220, 250,  162,  56,   8;\n  8: 1, 16,  84, 224, 350,  336, 196,  64,  9;\n  9: 2, 34, 192, 560, 980, 1092, 784, 352, 90, 10;\n","recurrence_sentence":"The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = -(6*n-10+(9*n^2-11*n-6)*x+(3*n^3-7*n)*x^2+(2*n^3-3*n^2-n+2)*x^3)*P(n-1,x) + (3+(6*n+4)*x+(3*n^2+9*n)*x^2+(5*n^2+3*n-2)*x^3+(2*n^2-1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 10 + (9*n^2 - 11*n - 6)*x + (3*n^3 - 7*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*P[n - 1, x] + (3 + (6*n + 4)*x + (3*n^2 + 9*n)*x^2 + (5*n^2 + 3*n - 2)*x^3 + (2*n^2 - 1)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n - 10 + (9*n^2 - 11*n - 6)*x + (3*n^3 - 7*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*v[i - 1] + (3 + (6*n + 4)*x + (3*n^2 + 9*n)*x^2 + (5*n^2 + 3*n - 2)*x^3 + (2*n^2 - 1)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*x^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-d75669a8cd7f9393","steps":["\u03b8+1"],"edges":["e-b7809f2be8630f6c"],"text":"A133087 \u2192 \u03b8+1"},"headline":"A133087 \u2192 \u03b8+1","chain":null}