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A263363
Expansion of Product_{k>=1} 1/(1-x^(k+7))^k.
8
1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 17, 22, 32, 42, 59, 76, 103, 130, 171, 216, 280, 354, 460, 584, 757, 968, 1249, 1596, 2056, 2618, 3354, 4266, 5441, 6900, 8778, 11108, 14094, 17814, 22546, 28450, 35946, 45280, 57088, 71806, 90347
OFFSET
0,10
LINKS
FORMULA
G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^k)^2).
a(n) ~ exp(1/12 - 49*Pi^4/(432*Zeta(3)) - 7*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^(269/36) * Pi^3 / (398131200 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(287/36)), 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-7), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Oct 16 2015
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 16 2015
STATUS
approved