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%I A294586 #16 Nov 14 2017 02:49:16
%S A294586 1,1,17,98,514,2435,12752,58849,277362,1243056,5523734,23889860,
%T A294586 102176581,427458488,1768064752,7197695011,28955246228,114977761216,
%U A294586 451686925462,1754581791860,6749143188662,25707194720502,97041994691555,363121143230292
%N A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).
%H A294586 Seiichi Manyama, <a href="/A294586/b294586.txt">Table of n, a(n) for n = 0..2889</a>
%F A294586 From _Vaclav Kotesovec_, Nov 14 2017: (Start)
%F A294586 a(n) ~ c * 3^(2*n/3) * n^8, where
%F A294586 if mod(n,3)=0 then c = 350793443467906700358779160929996923840677857044\
%F A294586 13786172.61998576944425459411592809123023259309183199454386580509531344\
%F A294586 26216683391121761062030679551011342614958936988089343473390138...
%F A294586 if mod(n,3)=1 then c = 350793443467906700358779160929996923840677857044\
%F A294586 13786172.61998576943431618172412821798685989333734080090574886961583670\
%F A294586 65437558779530384541992249698997443314123905740649930258416583...
%F A294586 if mod(n,3)=2 then c = 350793443467906700358779160929996923840677857044\
%F A294586 13786172.61998576943586440772541471067224229278174424709431922476448338\
%F A294586 37991534958575385658058309282842532811502400165735702386411333...
%F A294586 In closed form, a(n) ~ ((Product_{k>=4} ((1 - k^2 / 3^(2*k/3))^(-k^2))) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^4) + (Product_{k>=4} ((1 - (-1)^(2*k/3) * k^2 / 3^(2*k/3))^(-k^2))) / ((-1)^(2*n/3) * (1 + 4/3 * (-1/3)^(1/3))^4 * (1 - (-1/3)^(2/3))) + (Product_{k>=4} ((1 - (-(-1)^(1/3))^k * k^2 / 3^(2*k/3))^(-k^2))) / ((-(-1)^(1/3))^n * (1 + (-1)^(1/3) / 3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^4)) * 3^(2*n/3) * n^8 / 793618560. - _Vaclav Kotesovec_, Nov 14 2017 (End)
%t A294586 nmax = 30; CoefficientList[Series[Product[1/(1 - k^2*x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 11 2017 *)
%t A294586 nmax = 30; s = 1 - x; Do[s *= Sum[Binomial[k^2, j]*(-1)^j*k^(2*j)*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 12 2017 *)
%o A294586 (PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^2*x^k)^k^2))
%Y A294586 Column k=2 of A294585.
%Y A294586 Cf. A077335, A285241, A285674, A294584.
%K A294586 nonn
%O A294586 0,3
%A A294586 _Seiichi Manyama_, Nov 03 2017