A285461 - OEIS

%I #9 Apr 19 2017 10:33:22

%S 1,1,3,6,13,25,49,89,166,295,526,909,1571,2657,4475,7432,12257,20000,

%T 32436,52126,83285,132057,208221,326202,508372,787777,1214828,1863932,

%U 2847020,4328765,6554359,9882795,14843999,22210386,33112817,49192218,72834243

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

%C In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^(m*k)) / (1 - x^k))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (4 + 3/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (4 + 3/m^2)^(7/36) * Zeta(3)^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

%H Vaclav Kotesovec, <a href="/A285461/b285461.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (103*Zeta(3))^(1/3) * n^(2/3)) * (103*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(7/18) * sqrt(3*Pi) *  n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

%t nmax = 40; CoefficientList[Series[Product[((1+x^(5*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A156616 (m=1), A285462 (m=2), A285447 (m=3), A285460 (m=4).

%Y Cf. A024789.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Apr 19 2017