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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":"3f80c8faa73d9d792923562e972ff86b560522199503b17c8787097b82243516","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e91c3b4104e64a83","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (14 + x)^n.","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:          15,           4;\n  2:         225,         120,          12;\n  3:        3375,        2700,         540,          32;\n  4:       50625,       54000,       16200,        1920,         80;\n  5:      759375,     1012500,      405000,       72000,       6000,        192;\n  6:    11390625,    18225000,     9112500,     2160000,     270000,      17280,       448;\n  7:   170859375,   318937500,   191362500,    56700000,    9450000,     907200,     47040,    1024;\n  8:  2562890625,  5467500000,  3827250000,  1360800000,  283500000,   36288000,   2822400,  122880,   2304;\n  9: 38443359375, 92264062500, 73811250000, 30618000000, 7654500000, 1224720000, 127008000, 8294400, 311040, 5120;\n","recurrence_sentence":"The row polynomials satisfy (15+2*n*x)*P(n,x) = (225+(30*n+60)*x+(4*n+4)*x^2)*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[((225 + (30*n + 60)*x + (4*n + 4)*x^2)*P[n - 1, x])/(15 + 2*n*x)]];\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] = ((225 + (30*n + 60)*x + (4*n + 4)*x^2)*v[i - 1])/(15 + 2*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-a679863398df5985","steps":["basis A013609","\u03b8+1"],"edges":["e-267357ec4c02d3cf","e-fc407ad7d959a766"],"text":"A147716 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A147716 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}