A285263 - OEIS

%I #9 Apr 15 2017 19:30:39

%S 1,1,3,6,13,23,47,83,154,269,474,809,1387,2313,3859,6330,10341,16680,

%T 26790,42586,67375,105731,165097,256052,395248,606501,926502,1408048,

%U 2130788,3209643,4815595,7194875,10709843,15881236,23467805,34556842,50720003,74200845

%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(3 * 2^(-2/3) * ((1-1/m^2)*Zeta(3))^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(1/3) * sqrt(3*Pi) * m^(1/12) * n^(2/3)).

%H Vaclav Kotesovec, <a href="/A285263/b285263.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ exp(2^(1/3) * 3^(4/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/6) / (3^(1/3) * 5^(5/12) * sqrt(Pi) * n^(2/3)).

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

%Y Cf. A026007 (m=2), A263346 (m=3), A285262 (m=4).

%Y Cf. A285246.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Apr 15 2017