MacMahon-type $q$-series

Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we study a class of $q$-series arising from nested divisor structures. In particular, we consider the $q$-series $$V_k(q)=\sum _{1\le n_1\le n_2\le \cdots \le n_k}\frac{q^{n_1+n_2+\cdots +n_k}}{(1-q^{n_1}{)}^2(1-q^{n_2}{)}^2\cdots (1-q^{n_k}{)}^2},$$ introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series $$W_k(q)=\sum _{1\le n_1\le n_2\le \cdots \le n_k}\frac{q^{2(n_1+n_2+\cdots +n_k)-k}}{(1-q^{2n_1-1}{)}^2(1-q^{2n_2-1}{)}^2\cdots (1-q^{2n_k-1}{)}^2},$$ and establish families of identities, generating function relations, and hypergeometric representations for the truncated forms of $V_k(q)$ and $W_k(q)$. Connections with overpartition pairs and bipartitions with distinct odd parts arise naturally in this context.