A261384 Expansion of Product_{k>=1} (1+x^k)^(2*k-1) / (1-x^k)^(2*k).

1, 3, 12, 39, 117, 331, 893, 2307, 5766, 13986, 33046, 76302, 172567, 383013, 835731, 1795236, 3801105, 7941439, 16386777, 33423342, 67435311, 134675784, 266385932, 522135379, 1014643823, 1955656848, 3740191268, 7100290646, 13383997996, 25058666367

Offset: 0

Author

Vaclav Kotesovec, Aug 17 2015

Keywords

• nonn

Comments

Convolution of A161870 and A255835.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A006950, A015128, A156616, A161870, A255835, A261386.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k-1)/(1-x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (7*Zeta(3))^(2/9) * exp(1/6 - Pi^4/(6048*Zeta(3)) - Pi^2 * n^(1/3) / (12*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * 2^(1/6) * sqrt(3*Pi) * n^(13/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.