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%I #11 Aug 22 2018 10:34:05
%S 1,0,2,6,15,32,79,172,397,860,1879,3986,8462,17586,36408,74366,150875,
%T 303006,604511,1195872,2350614,4587484,8898857,17154278,32883109,
%U 62679852,118858190,224238730,421021209,786793776,1463796383,2711552690,5002097398,9190449808
%N Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)).
%H Vaclav Kotesovec, <a href="/A258348/b258348.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ 1 / (2^(3/2) * 15^(5/48) * Pi^(1/12) * n^(29/48)) * exp(-Zeta'(-1) - Zeta(3)/(4*Pi^2) - 75*Zeta(3)^3 / Pi^8 - 15^(5/4) * Zeta(3)^2 / (2*Pi^5) * n^(1/4) - sqrt(15) * Zeta(3) / Pi^2 * sqrt(n) + 4*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).
%F G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/k). - _Ilya Gutkovskiy_, Aug 22 2018
%t nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]
%t Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[3, k]-DivisorSigma[2, k])*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 100}] (* _Vaclav Kotesovec_, Apr 11 2016, following a suggestion of _George Beck_ *)
%Y Cf. A000294, A023871, A258344, A258347, A258349, A258350, A258351, A258352.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, May 27 2015
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