Compute the sum given a recursion for each summand

I have a system of two recursive equations with two unknowns A[n] and B[n] which apparently Mathematica can’t solve

RSolve not evaluating when given my system of recurrence equations

Each $ A[n]$ is a polynomial divided by a Pochhammer symbol: $ A[n] = p(q)/(q)_n = p(q)/\prod_{j=1}^n (1-q^j)$ where $ p(q) = \mathbb{Z}[q]$ .

I am only interested in computing the formal sum $ \sum_{n \geq 0} A[n]$ as a power series in $ q$ . Is this something Mathematica can do? I know how to get the corresponding polynomial to each order of $ n$ but I am interested in a closed formula for the formal sum.

For completeness I am copying the system of equations with initial conditions here:

(1-q^n)A[n] = q^{3n-1} B[n-1] + q^{2n+1} A[n-1] + q^{4n-2} (A[n-2]+B[n-2]) B[n] = q^{n+1} A[n-1] - q^{5n -5}(A[n-3]+B[n-3]) A[0] = 1, A[1]=q^3/(1-q), A[2] = q^6/(q)_2 B[0] = 0, B[1] = q^2, B[2] = q^6/(1-q)