Algorithmic determination of the enumerator for sums of three triangular numbers (Q2735228)

Let \(q(n)\), \(n\in \mathbb{N}\), be the number of partitions of \(n\) into distinct parts, \(q(0)=1\) and \(q(n)=0\) for \(n<0\), and \(t_3(n)\), \(n\in \mathbb{N}\), be the number of representations of \(n\) as a sum of three triangular numbers, i.e. \(n= \frac{h(h+1)}{2}+ \frac{j(j+1)}{2}+ \frac{k(k+1)}{2}\), \(h,j,k\in \mathbb{N}\). Then NEWLINE\[NEWLINEt_3(n)= q(n)- \sum_{k=1}^\infty (-1)^k q(n-3k^2+2k)(3k-1)+ \sum_{k=1}^\infty (-1)^k q(n-3k^2-2k)(3k+1).NEWLINE\]