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%I #21 Jul 15 2023 05:50:20
%S 0,1,2,-5,-28,85,806,-3185,-41656,207913,3428810,-20824925,-413027284,
%T 2961364861,68560259054,-567040692425,-15005357203312,140642298254929,
%U 4187120881320338,-43861384856264885,-1450918780756640140,16798626454194814117,611263061851828001462,-7751163512199032905505
%N Expansion of e.g.f. 3*x/(exp(-3*x)+exp(-x)+exp(x))
%C The terms of even indices are related to Bernoulli numbers. For example, 413027284 = 2^2 * 23 * 73 * 89 * 691 and 15005357203312 = 2^4 * 7 * 31 * 41 * 151 * 193 * 3617.
%C The terms of odd indices are related to the generalized Bernoulli numbers attached to the primitive Dirichlet character of period 3 (see A002111).
%F E.g.f.: 3*x/(exp(-3*x)+exp(-x)+exp(x)).
%o (Sage)
%o x = PowerSeriesRing(QQ, 'x').gen()
%o N = 20
%o f = (3*x/((-3*x).exp(N)+(-x).exp(N)+(x).exp(N))).egf_to_ogf()
%o print(list(f))
%o (PARI) my(N=25, x='x+O('x^N)); Vec(serlaplace(3*x/(exp(-3*x)+exp(-x)+exp(x))), -N) \\ _Michel Marcus_, Jul 13 2023
%Y Very similar to A083007.
%Y Related to A158073 and A002111.
%K sign
%O 0,3
%A _F. Chapoton_, Jul 13 2023
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