A337026 a(n) = (2/3) * Sum_{k>=0} (2*k + 1)^n / 3^k.

1, 2, 7, 38, 277, 2522, 27547, 351038, 5112457, 83764082, 1524907087, 30536665238, 667096092637, 15787642820042, 402374890155427, 10987722264846638, 320046586135452817, 9904844539648850402, 324568009210656076567, 11226512280285374623238

Offset: 0

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A080253, A122704, A123227.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 2*Exp(x)/(3-Exp(2*x)) ))); // G. C. Greubel, Jun 09 2022

Mathematica

Table[2^(n + 1) HurwitzLerchPhi[1/3, -n, 1/2]/3, {n, 0, 19}]
nmax = 19; CoefficientList[Series[2 Exp[x]/(3 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!

Sage

def A337026_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*exp(x)/(3-exp(2*x)) ).egf_to_ogf().list()
A337026_list(40) # G. C. Greubel, Jun 09 2022

Formula

E.g.f.: 2 * exp(x) / (3 - exp(2*x)).

a(n) = Sum_{k=0..n} binomial(n,k) * A122704(k).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A123227(k).

a(n) ~ n! * 2^(n+1) / (sqrt(3) * log(3)^(n+1)). - Vaclav Kotesovec, Mar 27 2022

a(n) = 1 + Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 24 2023