OFFSET
0,2
COMMENTS
In general, if g.f. = Product_{j>=1} (1+x^j)^(k^j), then a(n) ~ k^n * exp(2*sqrt(n) - 1/2 - c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} (-1)^m/(m*(k^(m-1)-1)).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of sequence A034691
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. 27.
FORMULA
a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .
MATHEMATICA
nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 16 2015
STATUS
approved