%I #9 Apr 09 2018 22:33:09
%S 1,2,12,46,175,610,2107,6918,22256,69498,212649,636910,1874470,
%T 5423332,15457223,43433088,120467606,330077358,894193347,2396636236,
%U 6359325300,16714566278,43539016461,112449776138,288083439729,732356943548,1848098069644,4630892393996
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)^2/2).
%C Euler transform of A006002.
%H Vaclav Kotesovec, <a href="/A302447/b302447.txt">Table of n, a(n) for n = 0..6000</a>
%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)^A006002(k).
%F a(n) ~ exp(5 * (3*Zeta(5))^(1/5) * n^(4/5) / 2^(8/5) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * (3*Zeta(5))^(3/5)) + (Zeta(3) / 2^(9/5) - Pi^8 / (27000 * 2^(4/5) * Zeta(5))) * n^(2/5) / (3*Zeta(5))^(2/5) + (Pi^8 / (12150000 * Zeta(5)) - Zeta(3) / 900) * Pi^4 * n^(1/5) / (2^(2/5) * 3^(1/5) * Zeta(5)^(6/5)) + 1/24 - Zeta(3) / (4*Pi^2) - Pi^16 / (5248800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (324000 * Zeta(5)^2) - Zeta(3)^2 / (120 * Zeta(5)) + Zeta'(-3)/2) * (3*Zeta(5))^(43/400) / (2^(57/200) * sqrt(5*A*Pi) * n^(243/400)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 08 2018
%t nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)^2/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]
%Y Cf. A000294, A006002, A294667.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 08 2018