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%I #10 Aug 28 2018 13:03:35
%S 1,1,0,0,0,0,0,0,8,8,0,0,0,0,0,0,28,28,0,0,0,0,0,0,56,56,0,27,27,0,0,
%T 0,70,70,0,216,216,0,0,0,56,56,0,756,756,0,0,0,28,28,0,1512,1512,0,
%U 351,351,8,8,0,1890,1890,0,2808,2808,65,65,0,1512,1512,0
%N Expansion of Product_{k>=1} (1+x^(k^3))^(k^3).
%C In general, if m > 0 and g.f. = Product_{k>=1} (1+x^(k^m))^(k^m), then a(n, m) ~ c * exp((2*m+1) * ((2^(1+1/m)-1) * Gamma(1/m) * Zeta(2+1/m))^(m/(2*m+1)) * n^((m+1)/(2*m+1)) / ((2*m+2)^((m+1)/(2*m+1)) * m^(3*m/(2*m+1)))) * ((2^(1+1/m)-1) * (m+1) * Gamma(1/m) * Zeta(2+1/m))^(m/(4*m+2)) / (sqrt(2*m+1) * sqrt(Pi) * 2^((3*m+2)/(4*m+2)) * m^((m-1)/(4*m+2)) * n^((3*m+1)/(4*m+2))), where c = 2^(-1/12) for m = 1 and c = 1 for m > 1.
%H Vaclav Kotesovec, <a href="/A291692/b291692.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ exp(7*((2^(4/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(12/7) * 3^(9/7))) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/14) / (2^(5/14) * 3^(1/7) * sqrt(7*Pi) * n^(5/7)).
%t nmax = 100; CoefficientList[Series[Product[(1 + x^(k^3))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^3, j]*x^(j*k^3), {j, 0, Floor[nmax/k^3] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax]
%Y Cf. A026007 (m=1), A291649 (m=2).
%Y Cf. A033461, A279329.
%K nonn
%O 0,9
%A _Vaclav Kotesovec_, Aug 30 2017
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