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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":"3390c68bab185b3bc8e76cc1e909f1d76a8220be4fe3fa5ba884a4a83fcde0ed","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-e03395d80bc5749d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(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:         11,           8;\n  2:        121,         176,          48;\n  3:       1331,        2904,        1584,         256;\n  4:      14641,       42592,       34848,       11264,        1280;\n  5:     161051,      585640,      638880,      309760,       70400,        6144;\n  6:    1771561,     7730448,    10541520,     6814720,     2323200,      405504,      28672;\n  7:   19487171,    99207416,   162339408,   131183360,    59628800,    15611904,    2207744,    131072;\n  8:  214358881,  1247178944,  2380977984,  2308827136,  1311833600,   457949184,   97140736,  11534336,   589824;\n  9: 2357947691, 15433839432, 33673831488, 38095647744, 25974305280, 11334242304, 3205644288, 570949632, 58392576, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (11+4*n*x)*P(n,x) = (121+(44*n+88)*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[((121 + (44*n + 88)*x + (16*n + 16)*x^2)*P[n - 1, x])/(11 + 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] = ((121 + (44*n + 88)*x + (16*n + 16)*x^2)*v[i - 1])/(11 + 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-a920b56c9b3ef4d1","steps":["basis A013611","\u03b8+1"],"edges":["e-551d764375152d3c","e-51bf225d74c9fcf7"],"text":"A038303 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038303 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}