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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"39287f2ab2130c3b125c495874eb44480ff44dd9078a60fa8409e857f28e29ff","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-16c98a2b049669c5","role":"sequence","name":"basis_map_A013610 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,            6;\n  2:         256,          192,           27;\n  3:        4096,         4608,         1296,          108;\n  4:       65536,        98304,        41472,         6912,         405;\n  5:     1048576,      1966080,      1105920,       276480,       32400,        1458;\n  6:    16777216,     37748736,     26542080,      8847360,     1555200,      139968,       5103;\n  7:   268435456,    704643072,    594542592,    247726080,    58060800,     7838208,     571536,     17496;\n  8:  4294967296,  12884901888,  12683575296,   6341787648,  1857945600,   334430208,   36578304,   2239488,   59049;\n  9: 68719476736, 231928233984, 260919263232, 152202903552, 53508833280, 12039487488, 1755758592, 161243136, 8503056, 196830;\n","recurrence_sentence":"The row polynomials satisfy (16+3*n*x)*P(n,x) = (256+(48*n+96)*x+(9*n+9)*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 + (48*n + 96)*x + (9*n + 9)*x^2)*P[n - 1, x])/(16 + 3*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 + (48*n + 96)*x + (9*n + 9)*x^2)*v[i - 1])/(16 + 3*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":["basis A013610","\u03b8+1"],"edges":["e-d2b5f968b40c2621","e-bf8fc077b8c19180"],"text":"A027467 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A027467 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}