A261998 Expansion of Product_{k>=1} (1-x^k)*(1+x^k)^4.

1, 3, 5, 10, 17, 26, 43, 65, 95, 140, 201, 283, 395, 545, 740, 1002, 1343, 1780, 2350, 3077, 4002, 5183, 6670, 8535, 10880, 13801, 17426, 21925, 27475, 34297, 42677, 52926, 65415, 80625, 99077, 121403, 148386, 180890, 219960, 266857, 323002, 390086, 470125

Offset: 0

Author

Vaclav Kotesovec, Sep 08 2015

Keywords

• nonn

Comments

In general, if m > 2 and g.f. = Product_{k>=1} (1-x^k)*(1+x^k)^m, then a(n) ~ exp(Pi*sqrt((m-2)*n/3)) / (2^((m+1)/2) * sqrt(n)).

Equals A000009 convolved with A085140. - George Beck, Jul 03 2016

Links

• 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. 16.

Crossrefs

Cf. A085140, A262380.

Programs

Mathematica

nmax = 80; CoefficientList[Series[Product[(1 - x^k) * (1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/2) * sqrt(n)).