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%I #7 Nov 29 2017 03:41:50
%S 1,2,15,58,235,862,3122,10664,35639,115164,363806,1122050,3393316,
%T 10068006,29374056,84347944,238713339,666419456,1836986443,5003473866,
%U 13476019215,35912177618,94746481999,247597696802,641205816641,1646268490598,4192059724668,10590937903412,26556243826240
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k*(2*k-1)).
%C Euler transform of A002939.
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^A002939(k).
%F a(n) ~ exp(2^(5/2) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) * sqrt(15*n) / Pi^2 - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(3/2) * Pi^5) - Zeta(3) / Pi^2 - 75*Zeta(3)^3 / (2*Pi^8) - 1/6) * A^2 / (2^(4/3) * 15^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Nov 28 2017
%t nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^(2 k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[2 d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
%Y Cf. A002939, A161870, A253289, A258347, A258348, A278767.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 28 2017
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