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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":"43c1e77059ae7fa6b4c1acddbf511abc518ce773076faa3ef0e354297dbee565","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-b203eb852c56fa2f","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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:         11,          40;\n  2:        121,         880,         1200;\n  3:       1331,       14520,        39600,         32000;\n  4:      14641,      212960,       871200,       1408000,         800000;\n  5:     161051,     2928200,     15972000,      38720000,       44000000,       19200000;\n  6:    1771561,    38652240,    263538000,     851840000,     1452000000,     1267200000,      448000000;\n  7:   19487171,   496037080,   4058485200,   16397920000,    37268000000,    48787200000,    34496000000,    10240000000;\n  8:  214358881,  6235894720,  59524449600,  288603392000,   819896000000,  1431091200000,  1517824000000,   901120000000,   230400000000;\n  9: 2357947691, 77169197160, 841845787200, 4761955968000, 16233940800000, 35419507200000, 50088192000000, 44605440000000, 22809600000000, 5120000000000;\n","recurrence_sentence":"The row polynomials satisfy (11+20*n*x)*P(n,x) = (121+(220*n+440)*x+(400*n+400)*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[((121 + (220*n + 440)*x + (400*n + 400)*x^2)*P[n - 1, x])/(11 + 20*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] = ((121 + (220*n + 440)*x + (400*n + 400)*x^2)*v[i - 1])/(11 + 20*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-e6ebe74518d42cbe","steps":["basis A013611","\u03b8+1"],"edges":["e-17e7dceca4b24148","e-b03e8fb7d7f565c0"],"text":"A038259 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038259 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}