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A329266 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 = 4.

%I A329266 #15 Nov 25 2019 04:53:52
%S A329266 1,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,4,0,0,0,0,6,0,0,0,0,0,0,4,0,
%T A329266 0,0,2,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,0,0,0,0,0,4,8,0,0,
%U A329266 4,0,0,0,0,4,0,0,0,0,0,0,8,4,4,0,0,4,0,0
%N A329266 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 = 4.
%H A329266 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 A329266 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 = 4 (see Proposition 1.1 in Zhou and Sun).
%e A329266 a(25) = 6 since there are 6 integer solutions to 1^4*k1^2 + 2^4*k2^2 + ... = 25:
%e A329266 k1 = +-5 and k_j = 0 for j > 1;
%e A329266 k1 = -3, k2 = +-1 and k_j = 0 for j > 2;
%e A329266 k1 = 3, k2 = +-1 and k_j = 0 for j > 2.
%t A329266 nmax=87;r=4;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 A329266 Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329265 (r = 3).
%K A329266 nonn
%O A329266 0,2
%A A329266 _Stefano Spezia_, Nov 09 2019