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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":"4bfeb8bb85e4eab9db5cd23ee70191aa5dbc529e85a963265b00a6dd7c9b98b6","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-ff0214c333bfdcca","role":"sequence","name":"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:          15,           4;\n  2:         225,         120,           9;\n  3:        3375,        2700,         405,          16;\n  4:       50625,       54000,       12150,         960,         25;\n  5:      759375,     1012500,      303750,       36000,       1875,        36;\n  6:    11390625,    18225000,     6834375,     1080000,      84375,      3240,       49;\n  7:   170859375,   318937500,   143521875,    28350000,    2953125,    170100,     5145,     64;\n  8:  2562890625,  5467500000,  2870437500,   680400000,   88593750,   6804000,   308700,   7680,    81;\n  9: 38443359375, 92264062500, 55358437500, 15309000000, 2392031250, 229635000, 13891500, 518400, 10935, 100;\n","recurrence_sentence":"The row polynomials satisfy (225+(45*n-15)*x+n^2*x^2)*P(n,x) = (3375+(675*n+675)*x+(15*n^2+75*n+45)*x^2+(n^2+2*n+1)*x^3)*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[((3375 + (675*n + 675)*x + (15*n^2 + 75*n + 45)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(225 + (45*n - 15)*x + n^2*x^2)]];\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] = ((3375 + (675*n + 675)*x + (15*n^2 + 75*n + 45)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(225 + (45*n - 15)*x + n^2*x^2)); 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":["(\u03b8+1)\u00b2"],"edges":["e-069f48e9a6ed6788"],"text":"A027467 \u2192 (\u03b8+1)\u00b2"},"headline":"A027467 \u2192 (\u03b8+1)\u00b2","chain":null}