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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":"b365e532e9af84745e37d9f5ca3a130a5c9c7b5034b6363db637dc40f4526332","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-5fccad4e67ef9278","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","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:     3;\n  2:     9,      8;\n  3:    27,     72;\n  4:    81,    432,      48;\n  5:   243,   2160,     720;\n  6:   729,   9720,    6480,    256;\n  7:  2187,  40824,   45360,   5376;\n  8:  6561, 163296,  272160,  64512,  1280;\n  9: 19683, 629856, 1469664, 580608, 34560;\n","recurrence_sentence":"The row polynomials satisfy -(81*n-81-(6*n^3-6*n)*x)*P(n,x) = -(243*n-243-(18*n^3-234*n+216)*x-(8*n^3+16*n^2-8*n-16)*x^2)*P(n-1,x) - (486*x-(36*n^2+108*n+288)*x^2+(16*n^2+48*n+32)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(243*n - 243 - (18*n^3 - 234*n + 216)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (486*x - (36*n^2 + 108*n + 288)*x^2 + (16*n^2 + 48*n + 32)*x^3)*D[P[n - 1, x], x])/(-81*n + 81 + (6*n^3 - 6*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(243*n - 243 - (18*n^3 - 234*n + 216)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*v[i - 1] - (486*x - (36*n^2 + 108*n + 288)*x^2 + (16*n^2 + 48*n + 32)*x^3)*deriv(v[i - 1], x))/(-81*n + 81 + (6*n^3 - 6*n)*x)); 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-01cea33b6485d9cc","steps":["Veronese 2","\u03b8+1"],"edges":["e-4fd32b1a93a66d21","e-de9f63aa4ee8a7a3"],"text":"A038220 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A038220 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}