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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":"730294a3d4f52a6511b93fe31d7e53b1a41ca89c732a4940e94dfc031b71a7be","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-39736a4ef132a31d","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:       6,        4;\n  2:      35,       48,        12;\n  3:     204,      424,       216,        32;\n  4:    1189,     3312,      2556,       768,       80;\n  5:    6930,    24204,     25056,     11392,     2400,      192;\n  6:   40391,   169632,    220392,    134400,    42800,     6912,     448;\n  7:  235416,  1155152,   1806192,   1382720,   590400,   144000,   18816,    1024;\n  8: 1372105,  7702944,  14082216,  12978432,  6956400,  2274048,  448448,   49152,   2304;\n  9: 7997214, 50550932, 105795936, 114040448, 73594080, 30195648, 7977984, 1318912, 124416, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(16*n+3*n*x-n^3*x^2)*P(n,x) = -(48*n+48+(8*n^2+41*n+33)*x-(3*n^3+3*n^2-6*n-6)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (64+72*x-(8*n^2+8*n-26)*x^2-(6*n^2+6*n-3)*x^3-(n^2+n)*x^4)*d/dx 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[(-(48*n + 48 + (8*n^2 + 41*n + 33)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (64 + 72*x - (8*n^2 + 8*n - 26)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-16*n - 3*n*x + n^3*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] = (-(48*n + 48 + (8*n^2 + 41*n + 33)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (64 + 72*x - (8*n^2 + 8*n - 26)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-16*n - 3*n*x + n^3*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-c2b7c74f8491f01b","steps":["basis A013609","\u03b8+1"],"edges":["e-63a398c0ccd31fef","e-319242eff2732a2b"],"text":"A207824 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A207824 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}