Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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