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%I #6 Apr 09 2018 22:33:21
%S 1,1,11,46,185,700,2676,9646,34166,117500,396506,1310527,4258313,
%T 13607309,42846151,133039791,407833188,1235202869,3699140386,
%U 10960888382,32154531807,93437164720,269087234273,768340525743,2176098269286,6115444177489,17058887661133
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(4*k^2-1)/3).
%C Euler transform of A000447.
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/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)^A000447(k).
%F a(n) ~ exp(5 * Zeta(5)^(1/5) * n^(4/5)/2 - Zeta(3) * n^(2/5) / (12 * Zeta(5)^(2/5)) + 4*Zeta'(-3)/3 - 1/36 - Zeta(3)^2 / (720*Zeta(5))) * A^(1/3) * Zeta(5)^(83/900) / (2^(7/180) * sqrt(5*Pi) * n^(533/900)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 08 2018
%t nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^(k (4 k^2 - 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (4 d^2 - 1)/3, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]
%Y Cf. A000335, A000447, A023871, A279215.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 08 2018
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