OFFSET
0,9
LINKS
Vaclav Kotesovec, 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
FORMULA
G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(5*j))^2).
a(n) ~ Zeta(3)^(151/900) * exp(d52 - Pi^4/(2700*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (150 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(299/900) * 5^(151/450) * sqrt(3*Pi) * n^(601/900)), where d52 = A263179 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 2/(25*x^2) + 1/(300*x*exp(x)) = -0.187803021063745858976409657887070138806... .
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
`if`(irem(d+5, 5, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..100); # after Alois P. Heinz
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 10 2015
STATUS
approved