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A329264 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 = 2.

%I A329264 #18 Nov 25 2019 04:53:33
%S A329264 1,2,0,0,4,4,0,0,4,4,4,0,0,12,8,0,6,16,4,0,16,8,8,0,8,24,20,0,0,52,24,
%T A329264 0,12,32,28,8,24,12,48,16,24,68,48,8,16,96,32,16,8,68,96,32,40,68,128,
%U A329264 32,80,88,76,48,32,156,104,64,8,224,192,40,88,152,208
%N A329264 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 = 2.
%H A329264 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 A329264 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 = 2 (see Proposition 1.1 in Zhou and Sun).
%e A329264 a(16) = 6 since there are 6 integer solutions to 1^2*k1^2 + 2^2*k2^2 + 3^2*k3^2 + 4^2*k4^2 + ... = 16:
%e A329264 k1 = +-4 and k_j = 0 for j > 1;
%e A329264 k1 = 0, k2 = +-2 and k_j = 0 for j > 2;
%e A329264 k1 = k2 = k3 = 0, k4 = +-1 and k_j = 0 for j > 4.
%t A329264 nmax=70; r=2; 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 A329264 Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329265 (r = 3), A329266 (r = 4).
%K A329264 nonn
%O A329264 0,2
%A A329264 _Stefano Spezia_, Nov 09 2019