  Here we consider the $q$-series coming from the Hall-Littlewood polynomials,
\begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq
a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big).
\end{equation*} These series were defined by Griffin, Ono, and Warnaar in their
work on the framework of the Rogers-Ramanujan identities. We devise a recursive
method for computing the coefficients of these series when they arise within
the Rogers-Ramanujan framework. Furthermore, we study the congruence properties
of certain quotients and products of these series, generalizing the famous
Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}
