A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535

Offset: 0

Author

Vaclav Kotesovec, Oct 15 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• 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

Crossrefs

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.

Cf. A285289, A285290, A285291.

Programs

Mathematica

nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ Zeta(3)^(1/6) * exp(2^(-1/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(2/3) * sqrt(Pi) * n^(2/3)).