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%I #15 Nov 25 2019 04:53:42
%S 1,2,0,0,2,0,0,0,2,6,0,0,4,0,0,0,2,4,0,0,0,0,0,0,4,2,0,2,4,0,0,4,2,8,
%T 0,4,18,0,0,8,0,4,0,4,12,0,0,0,4,2,0,8,4,0,0,0,0,8,0,4,16,0,0,12,4,4,
%U 0,0,16,0,0,8,10,16,0,8,16,0,0,0,4,18,0,0,16
%N a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 3.
%H Nian Hong Zhou, Yalin Sun, <a href="https://arxiv.org/abs/1910.07884">Counting the number of solutions to certain infinite Diophantine equations</a>, arXiv:1910.07884 [math.NT], 2019.
%F a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 3 (see Proposition 1.1 in Zhou and Sun).
%e a(9) = 6 since there are 6 integer solutions to 1^3*k1^2 + 2^3*k2^2 + ... = 9:
%e k1 = +-3 and k_j = 0 for j > 1;
%e k1 = -1, k2 = +-1 and k_j = 0 for j > 2;
%e k1 = 1, k2 = +-1 and k_j = 0 for j > 2.
%t nmax=85; r=3; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]
%Y Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329266 (r = 4).
%K nonn
%O 0,2
%A _Stefano Spezia_, Nov 09 2019
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