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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":"83ac2147268d4b73470804f66b09204aff263912514cb01b546345d821de3a6f","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-0088aa50f9dba3f3","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^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:          13,           8;\n  2:         169,         208,           48;\n  3:        2197,        4056,         1872,          256;\n  4:       28561,       70304,        48672,        13312,        1280;\n  5:      371293,     1142440,      1054560,       432640,       83200,        6144;\n  6:     4826809,    17822064,     20563920,     11248640,     3244800,      479232,      28672;\n  7:    62748517,   270301304,    374263344,    255906560,    98425600,    21805056,    2609152,    131072;\n  8:   815730721,  4015905088,   6487231296,   5322856448,  2559065600,   755908608,  135675904,  13631488,   589824;\n  9: 10604499373, 58732611912, 108429437376, 103795700736, 59882135040, 22110326784, 5291360256, 797442048, 69009408, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (13+4*n*x)*P(n,x) = (169+(52*n+104)*x+(16*n+16)*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[((169 + (52*n + 104)*x + (16*n + 16)*x^2)*P[n - 1, x])/(13 + 4*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] = ((169 + (52*n + 104)*x + (16*n + 16)*x^2)*v[i - 1])/(13 + 4*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-81e6fb4f9f226d77","steps":["basis A013611","\u03b8+1"],"edges":["e-950a320de9db6fb9","e-298a83b947894419"],"text":"A038327 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038327 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}