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Expansion of Product_{k>=1} (1 + x^(k^2))^2.
8

%I #9 Dec 09 2016 17:41:10

%S 1,2,1,0,2,4,2,0,1,4,5,2,0,4,8,4,2,6,7,4,5,8,6,4,4,10,15,8,1,12,24,12,

%T 1,8,19,18,10,8,16,24,17,16,23,20,12,22,34,20,8,20,42,38,18,18,42,52,

%U 30,20,34,46,34,30,46,48,36,46,72,58,33,42,71,72,41

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

%H Vaclav Kotesovec, <a href="/A279226/b279226.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3) / 2) * sqrt(2/3) * ((sqrt(2)-1) * Zeta(3/2) / Pi)^(1/3) / (4*n^(5/6)). - _Vaclav Kotesovec_, Dec 09 2016

%t nmax = 100; CoefficientList[Series[Product[(1 + x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {p, 1, 2}], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Dec 09 2016 *)

%Y Cf. A033461, A103265, A279225, A279227.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 08 2016