A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).
1, 1, 9, 36, 140, 481, 1774, 5925, 20076, 64980, 208486, 652058, 2017023, 6117878, 18347256, 54222195, 158463794, 457570786, 1307951914, 3700153918, 10371860026, 28810051738, 79359812567, 216834266612, 587961817595, 1582612248239, 4230325722508
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Programs
-
Mathematica
nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]
Formula
a(n) ~ c * n^8 * 3^(n/3), where
if mod(n,3) = 0 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560142\
40331306860864399770618296475558098172993864629247911801570500913143642\
65158886200452165335605783203726486071335...
if mod(n,3) = 1 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560112\
77299895134841028015999951571187798033179513268954711586617617334007687\
07198348808962592621276659532114355538024...
if mod(n,3) = 2 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560117\
00278534968233203470801053870003971422069097966617636511346003845666735\
79293861331368526745743422198017148868212...
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).