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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":"a23af043b52b3462e22876249019343befaf15c50f17c382d8cd676f9cdd0b77","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-c20d7dca3713c0eb","role":"sequence","name":"basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:          16,           2;\n  2:         256,          64,           3;\n  3:        4096,        1536,         144,          4;\n  4:       65536,       32768,        4608,        256,         5;\n  5:     1048576,      655360,      122880,      10240,       400,        6;\n  6:    16777216,    12582912,     2949120,     327680,     19200,      576,       7;\n  7:   268435456,   234881024,    66060288,    9175040,    716800,    32256,     784,     8;\n  8:  4294967296,  4294967296,  1409286144,  234881024,  22937600,  1376256,   50176,  1024,    9;\n  9: 68719476736, 77309411328, 28991029248, 5637144576, 660602880, 49545216, 2408448, 73728, 1296, 10;\n","recurrence_sentence":"The row polynomials satisfy (16+n*x)*P(n,x) = (256+(16*n+32)*x+(n+1)*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[((256 + (16*n + 32)*x + (n + 1)*x^2)*P[n - 1, x])/(16 + 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] = ((256 + (16*n + 32)*x + (n + 1)*x^2)*v[i - 1])/(16 + 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-ff0214c333bfdcca","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-5b39e32205b8abe3","e-9ee5b6150c264ae1"],"text":"A027467 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A027467 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}