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Expansion of Product_{k>=1} (1 + x^(2*k^2 + 1)).
2

%I #6 Aug 31 2017 06:22:58

%S 1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,1,

%T 0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,

%U 0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,1,1,0

%N Expansion of Product_{k>=1} (1 + x^(2*k^2 + 1)).

%H Vaclav Kotesovec, <a href="/A291749/b291749.txt">Table of n, a(n) for n = 0..100000</a>

%F a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3)/4) * ((sqrt(2)-1) * Zeta(3/2))^(1/3) / (2 * sqrt(6) * Pi^(1/3) * n^(5/6)).

%t nmax = 200; CoefficientList[Series[Product[(1 + x^(2*k^2 + 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[4]] = 1; Do[Do[poly[[j + 1]] += poly[[j - 2*k^2]], {j, nmax, 2*k^2 + 1, -1}];, {k, 2, Sqrt[(nmax - 1)/2] + 1}]; poly

%Y Cf. A033461, A291748.

%K nonn

%O 0,74

%A _Vaclav Kotesovec_, Aug 31 2017