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Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k^2+1)/2).
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%I #6 Apr 09 2018 22:33:15

%S 1,1,6,21,70,210,646,1881,5446,15295,42355,115036,308312,814023,

%T 2123431,5471967,13949888,35194914,87952796,217803302,534794576,

%U 1302545064,3148316746,7554386885,18001627175,42613759083,100240372671,234371794954,544812235887

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k^2+1)/2).

%C Euler transform of A006003.

%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)^A006003(k).

%F a(n) ~ exp(5 * (3*Zeta(5))^(1/5) * n^(4/5) / 2^(8/5) + Zeta(3) * n^(2/5) / (2^(9/5) * (3*Zeta(5))^(2/5)) + Zeta'(-3)/2 + 1/24 - Zeta(3)^2 / (120 * Zeta(5))) * (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 = 28; CoefficientList[Series[Product[1/(1 - x^k)^(k (k^2 + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d^2 + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

%Y Cf. A000294, A006003, A295179, A302447.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 08 2018