A364199 Expansion of e.g.f. 2*x/(exp(-2*x)+exp(x)).

0, 1, 1, -6, -13, 110, 363, -4214, -18581, 276678, 1525355, -27753022, -183611829, 3948004606, 30473073547, -756031185030, -6669149100757, 187521633674294, 1860949703300139, -58481734930175438, -644853406058229365, 22398157925324204142, 271672536688626976331, -10334883450918076967446

Offset: 0

Author

F. Chapoton, Jul 13 2023

Keywords

• sign

Comments

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.

The terms with odd indices are related to the generalized Bernoulli numbers attached to the primitive Dirichlet character of period 3 (see A002111).

Crossrefs

Very similar to the Genocchi numbers A036968.

Related to A156179 and A002111.

Programs

PARI

my(N=25, x='x+O('x^N)); Vec(serlaplace(2*x/(exp(-2*x)+exp(x))), -N) \\ Michel Marcus, Jul 15 2023

Sage

x = PowerSeriesRing(QQ, 'x').gen()
N = 20
f = (2*x/((-2*x).exp(N)+(x).exp(N))).egf_to_ogf()
print(list(f))

Formula

E.g.f.: 2*x/(exp(-2*x)+exp(x)).