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A285048 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^(4*k+1). 4

%I

%S 1,1,1,1,1,6,6,6,6,15,30,30,30,43,88,123,123,140,250,385,455,476,678,

%T 1098,1413,1564,1913,2918,4048,4707,5452,7572,10747,13265,15195,19534,

%U 27349,35146,41042,50011,67596,88897,106519,126635,164230,216862,266473,314883

%N Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^(4*k+1).

%H Seiichi Manyama, <a href="/A285048/b285048.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from Vaclav Kotesovec)

%F a(n) ~ 4 * Pi * 2^(25/72) * Zeta(3)^(11/72) * exp(4*c + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/4)^3 * n^(47/72)), where c = Integral_{x=0..inf} ((-19/(exp(x)*96) + 1/(exp(x)*(1 - exp(-4*x))^2) - 1/(16*x^2) - 3/(16*x))/x) dx = 0.09601010361866957956805888476415949391295401812706635... - _Vaclav Kotesovec_, Apr 16 2017

%t nmax = 50; CoefficientList[Series[Product[1/(1-x^(4*k-3))^(4*k-3), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Apr 16 2017 *)

%Y Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A262811 (m=2), A262947 (m=3), this sequence (m=4), A285049 (m=5).

%Y Cf. A285070, A285287.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Apr 15 2017

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Last modified August 2 20:38 EDT 2021. Contains 346428 sequences. (Running on oeis4.)