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%I #16 Apr 24 2017 12:44:29
%S 1,1,9,36,140,481,1774,5925,20076,64980,208486,652058,2017023,6117878,
%T 18347256,54222195,158463794,457570786,1307951914,3700153918,
%U 10371860026,28810051738,79359812567,216834266612,587961817595,1582612248239,4230325722508
%N Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).
%H Vaclav Kotesovec, <a href="/A285241/b285241.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) ~ c * n^8 * 3^(n/3), where
%F if mod(n,3) = 0 then c = 3435237242728465092737309192093188152686332293\
%F 03276380306112638865540880372901642880694943679256417087889777743957063\
%F 209444405157397505005623042846150296486667845382334521513094023.8560142\
%F 40331306860864399770618296475558098172993864629247911801570500913143642\
%F 65158886200452165335605783203726486071335...
%F if mod(n,3) = 1 then c = 3435237242728465092737309192093188152686332293\
%F 03276380306112638865540880372901642880694943679256417087889777743957063\
%F 209444405157397505005623042846150296486667845382334521513094023.8560112\
%F 77299895134841028015999951571187798033179513268954711586617617334007687\
%F 07198348808962592621276659532114355538024...
%F if mod(n,3) = 2 then c = 3435237242728465092737309192093188152686332293\
%F 03276380306112638865540880372901642880694943679256417087889777743957063\
%F 209444405157397505005623042846150296486667845382334521513094023.8560117\
%F 00278534968233203470801053870003971422069097966617636511346003845666735\
%F 79293861331368526745743422198017148868212...
%F In closed form, a(n) ~ -(27*Product_{k>=4}((1 - k / 3^(k/3))^(-k^2)) / (13 + 128*3^(1/3) - 95*3^(2/3)) + 243*Product_{k>=4}((1 + (-1)^(1 + 2*k/3) * k / 3^(k/3))^(-k^2)) / ((-1)^(2*n/3) * ((3 + 2*(-3)^(1/3))^4 * (-3 + (-3)^(2/3)))) + (-1)^(1 - 4*n/3) * Product_{k>=4}((1 + (-1)^(1 + 4*k/3) * k / 3^(k/3))^(-k^2)) / ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^4)) / 793618560 * n^8 * 3^(n/3).
%t nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]
%Y Cf. A006906, A023871, A266941, A285240, A285243.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Apr 15 2017
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