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A261389 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k). 6

%I

%S 1,6,30,128,486,1704,5604,17484,52206,150118,417696,1128984,2973476,

%T 7650720,19272432,47616568,115570014,275921460,648771802,1503889488,

%U 3439990344,7770915816,17349229908,38306180052,83694778556,181052778078,387976101432,823939048560

%N Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k).

%C Convolution of A255610 and A027346.

%C In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(t*k) and t>=1, then a(n) ~ exp(t/12 + 3/2 * (7*t*Zeta(3)/2)^(1/3) * n^(2/3)) * t^(1/6 + t/36) * (7*Zeta(3))^(1/6 + t/36) / (A^t * 2^(2/3 + t/9) * sqrt(3*Pi) * n^(2/3 + t/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

%H G. C. Greubel, <a href="/A261389/b261389.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.

%F a(n) ~ exp(1/4 + 3/2 * (21*Zeta(3)/2)^(1/3) * n^(2/3)) * (7*Zeta(3)/3)^(1/4) / (2 * A^3 * sqrt(Pi) * n^(3/4)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

%t nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A156616 (t=1), A261386 (t=2).

%Y Cf. A015128, A027346, A027906, A193427, A255610.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 17 2015

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Last modified July 25 03:53 EDT 2021. Contains 346283 sequences. (Running on oeis4.)