Further study on MacMahon-type sums of divisors

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