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A263361
Expansion of Product_{k>=1} 1/(1-x^(k+5))^k.
8
1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 15, 20, 30, 40, 58, 76, 106, 140, 191, 252, 344, 454, 613, 814, 1091, 1442, 1926, 2538, 3368, 4432, 5852, 7678, 10107, 13222, 17337, 22636, 29582, 38518, 50195, 65198, 84712, 109784, 142254, 183924, 237742, 306688
OFFSET
0,8
LINKS
FORMULA
G.f.: exp(Sum_{k>=1} x^(6*k)/(k*(1-x^k)^2).
a(n) ~ exp(1/12 - 25*Pi^4/(432*Zeta(3)) - 5*Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(125/36) * Pi^2 / (576 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(143/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
max(0, d-5), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..50); # Alois P. Heinz, Oct 16 2015
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(6*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 16 2015
STATUS
approved