OFFSET
0,5
COMMENTS
a(n) is the number of partitions of n into parts 3*k-2 of k kinds (k>=1). - Joerg Arndt, Oct 06 2015
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
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)
FORMULA
a(n) ~ Zeta(3)^(13/108) * exp(d2 - Pi^4 / (972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.0145374291832840336050202945022620903605414975934644413815... . - Vaclav Kotesovec, Oct 08 2015
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
`if`(irem(d+3, 3, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Oct 05 2015
MATHEMATICA
nmax=100; CoefficientList[Series[Product[1/(1-x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[1/j*x^j/(1-x^(3j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 04 2015
STATUS
approved