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%I #16 Feb 04 2021 14:42:10
%S 1,2,0,0,2,4,0,0,0,2,4,0,0,4,8,0,2,4,0,0,4,8,0,0,0,6,12,0,0,12,24,0,0,
%T 0,4,8,2,4,8,16,4,12,8,0,0,12,24,0,0,10,28,16,4,12,24,32,8,16,4,8,0,
%U 12,32,16,2,32,56,0,4,16,24,16,0,4,36,56,0,16
%N Expansion of Product_{k>=1} (1+2*x^(k^2)).
%H Vaclav Kotesovec, <a href="/A279360/b279360.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ c^(1/3) * exp(3 * 2^(-4/3) * c^(2/3) * Pi^(1/3) * n^(1/3)) / (3 * 2^(2/3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -2) = 1.28138038315976963883198... . - _Vaclav Kotesovec_, Dec 12 2016
%F From _Alois P. Heinz_, Feb 03 2021: (Start)
%F a(n) = Sum_{k>=0} 2^k * A341040(n,k).
%F a(n) = 0 <=> n in { A001422 }. (End)
%t nmax = 200; CoefficientList[Series[Product[(1+2*x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 0; Do[Do[poly[[j + 1]] += 2*poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]
%Y Cf. A001422, A032302, A033461, A279226, A279368, A341040.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Dec 10 2016
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