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%I #12 Jul 15 2023 05:51:28
%S 0,1,1,-6,-13,110,363,-4214,-18581,276678,1525355,-27753022,
%T -183611829,3948004606,30473073547,-756031185030,-6669149100757,
%U 187521633674294,1860949703300139,-58481734930175438,-644853406058229365,22398157925324204142,271672536688626976331,-10334883450918076967446
%N Expansion of e.g.f. 2*x/(exp(-2*x)+exp(x)).
%C The terms with even indices are related to Bernoulli numbers. For example, 183611829 = 3 * 23 * 691 * 3851 and 6669149100757 = 11^2 * 13 * 257 * 3617 * 4561.
%C The terms with odd indices are related to the generalized Bernoulli numbers attached to the primitive Dirichlet character of period 3 (see A002111).
%F E.g.f.: 2*x/(exp(-2*x)+exp(x)).
%o (Sage)
%o x = PowerSeriesRing(QQ, 'x').gen()
%o N = 20
%o f = (2*x/((-2*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(2*x/(exp(-2*x)+exp(x))), -N) \\ _Michel Marcus_, Jul 15 2023
%Y Very similar to the Genocchi numbers A036968.
%Y Related to A156179 and A002111.
%K sign
%O 0,4
%A _F. Chapoton_, Jul 13 2023