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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":"6a1280d5df67783c5c2da0f3a9ef767a85c5f5a9220d56207093d9941cbf0b70","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-5d39164c4108e043","role":"sequence","name":"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: 0, 6;\n   3: 0, 2, 27;\n   4: 0, 0, 18, 108;\n   5: 0, 0,  3, 108, 405;\n   6: 0, 0,  0,  36, 540, 1458;\n   7: 0, 0,  0,   4, 270, 2430,  5103;\n   8: 0, 0,  0,   0,  60, 1620, 10206, 17496;\n   9: 0, 0,  0,   0,   5,  540,  8505, 40824,  59049;\n  10: 0, 0,  0,   0,   0,   90,  3780, 40824, 157464, 196830;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3)*P(n,x) = -(2*n-4-18*x)*P(n-1,x) + (4*x+9*x^2)*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[(-(2*n - 4 - 18*x)*P[n - 1, x] + (4*x + 9*x^2)*D[P[n - 1, x], x])/(3*n - 3)]];\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] = (-(2*n - 4 - 18*x)*v[i - 1] + (4*x + 9*x^2)*deriv(v[i - 1], x))/(3*n - 3)); 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":["\u03b8+1"],"edges":["e-a7f2bd7105b88589"],"text":"A099097 \u2192 \u03b8+1"},"headline":"A099097 \u2192 \u03b8+1","chain":null}