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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":"3161ed007c906ec4a2a5bc55460e87d1a35f3035a329e0093693cbc895bd41b2","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-f562ab3e3b669a1b","role":"sequence","name":"basis_map_A013610 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,          30;\n  2:        121,         660,          675;\n  3:       1331,       10890,        22275,         13500;\n  4:      14641,      159720,       490050,        594000,        253125;\n  5:     161051,     2196150,      8984250,      16335000,      13921875,       4556250;\n  6:    1771561,    28989180,    148240125,     359370000,     459421875,     300712500,      79734375;\n  7:   19487171,   372027810,   2282897925,    6917872500,   11791828125,   11577431250,    6139546875,    1366875000;\n  8:  214358881,  4676921040,  33482502900,  121754556000,  259420218750,  339604650000,  270140062500,  120285000000,   23066015625;\n  9: 2357947691, 57876897870, 473538255300, 2008950174000, 5136520331250, 8405215087500, 8914622062500, 5954107500000, 2283535546875, 384433593750;\n","recurrence_sentence":"The row polynomials satisfy (11+15*n*x)*P(n,x) = (121+(165*n+330)*x+(225*n+225)*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 + (165*n + 330)*x + (225*n + 225)*x^2)*P[n - 1, x])/(11 + 15*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 + (165*n + 330)*x + (225*n + 225)*x^2)*v[i - 1])/(11 + 15*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 A013610","\u03b8+1"],"edges":["e-aef71e51dba5316b","e-ca4529b30be6ff69"],"text":"A038259 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038259 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}