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%I #7 Aug 28 2015 05:09:46
%S 1,6,18,38,66,108,182,306,486,728,1068,1578,2318,3312,4614,6388,8862,
%T 12192,16488,22038,29400,39156,51702,67554,87810,113982,147384,189200,
%U 241446,307356,390408,493662,621006,778712,974628,1216284,1511756,1872840,2315538
%N Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.
%C In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).
%F a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).
%t nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]
%Y Cf. A015128 (a=1, b=1, j=1), A156616.
%Y Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
%Y Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
%Y Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Aug 28 2015
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