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A263406
Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.
6
1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 2, 4, 1, 6, 5, 2, 10, 7, 6, 19, 9, 14, 29, 14, 28, 46, 23, 53, 66, 43, 95, 99, 76, 158, 143, 141, 256, 217, 247, 403, 326, 432, 617, 509, 720, 935, 801, 1187, 1399, 1281, 1892, 2087, 2047, 2983, 3107, 3272, 4589, 4647
OFFSET
0,9
LINKS
FORMULA
G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^(3*k))^2).
a(n) ~ c * Zeta(3)^(13/108) * exp(-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 c = 3^(1/3) * Gamma(1/3) * exp(A263030) / sqrt(2*Pi) = 1.2763162741536982965216627321306598385267089489...
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
`if`(irem(d-2, 3)=0, (d-2)/3, 0),
d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Oct 17 2015
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 17 2015
STATUS
approved