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A294586
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Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).
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3
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1, 1, 17, 98, 514, 2435, 12752, 58849, 277362, 1243056, 5523734, 23889860, 102176581, 427458488, 1768064752, 7197695011, 28955246228, 114977761216, 451686925462, 1754581791860, 6749143188662, 25707194720502, 97041994691555, 363121143230292
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * 3^(2*n/3) * n^8, where
if mod(n,3)=0 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576944425459411592809123023259309183199454386580509531344\
26216683391121761062030679551011342614958936988089343473390138...
if mod(n,3)=1 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943431618172412821798685989333734080090574886961583670\
65437558779530384541992249698997443314123905740649930258416583...
if mod(n,3)=2 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943586440772541471067224229278174424709431922476448338\
37991534958575385658058309282842532811502400165735702386411333...
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)
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MATHEMATICA
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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 *)
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 *)
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PROG
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(PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^2*x^k)^k^2))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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