A261386 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k).

1, 4, 16, 56, 176, 520, 1456, 3896, 10048, 25100, 60960, 144440, 334752, 760456, 1696464, 3722224, 8043040, 17135624, 36031104, 74840568, 153680928, 312198160, 627828272, 1250540024, 2468443296, 4830809868, 9377190336, 18061370288, 34531009760, 65552873736

Offset: 0

Author

Vaclav Kotesovec, Aug 17 2015

Keywords

• nonn

Comments

Convolution of A161870 and A026011.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.

Crossrefs

Cf. A015128, A026011, A156616, A161870, A216406, A261384, A261389.

Programs

Mathematica

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

Formula

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

G.f.: exp(Sum_{k>=1} (sigma_2(2*k) - sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Apr 14 2019