Further study on MacMahon-type sums of divisors

This paper is devoted to the study of \(\begin{aligned} U_t(a,q):=\sum _{1\le n_1<n_2<\cdots <n_t}\frac{q^{n_1+n_2+\cdots +n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots (1+aq^{n_t}+q^{2n_t})} \end{aligned}\) $$\begin{array}{c}\hfill U_t(a,q):=\sum _{1\le n_1<n_2<\cdots <n_t}\frac{q^{n_1+n_2+\cdots +n_t}}{(1+a{q}^{n_1}+q^{2n_1})(1+a{q}^{n_2}+q^{2n_2})\cdots (1+a{q}^{n_t}+q^{2n_t})}\end{array}$$ when a is one of \(0, \pm 1, \pm 2\) $0,\pm 1,\pm 2$ . The idea builds on our previous treatment of the case \(a=-2\) $a=-2$ . It is shown that all these functions lie in the ring of quasimodular forms. Among the more surprising findings is \(\begin{aligned} U_2(1,q)=\sum _{n\ge 1} \frac{q^{3n}}{(1-q^{3n})^2}. \end{aligned}\) $$\begin{array}{c}\hfill U_2(1,q)=\sum _{n\ge 1}\frac{q^{3n}}{{(1-q^{3n})}^2}.\end{array}$$