A285291 - OEIS

%I #9 Apr 16 2017 06:15:37

%S 1,1,2,5,8,15,27,47,78,134,218,356,576,916,1449,2268,3525,5431,8324,

%T 12652,19129,28754,42974,63898,94553,139241,204144,298045,433328,

%U 627592,905560,1301934,1865362,2663816,3791813,5380911,7613286,10740839,15111141,21202615

%N Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(5*k)))^k.

%C In general, if m > 1 and g.f. = Product_{k>=1} ((1 + x^k) / (1 + x^(m*k)))^k, then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * (1-1/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

%H Vaclav Kotesovec, <a href="/A285291/b285291.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ exp(2^(-1/3) * 3^(5/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (5^(1/3) * 6^(1/6) * sqrt(Pi) * n^(2/3)).

%t nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1+x^(5*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A285289 (m=2), A263345 (m=3), A285290 (m=4).

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Apr 16 2017