I'm trying to solve a recurrence relation, which looks roughly like $$a_{(n+1)(k+1)}=f(n,k)a_{n(k+1)}+g(n,k)(a_{nk}-a_{(n-k)k})$$, where the exact form of $c_{n,k},d{n,k}$ is some fraction of polynomials in $n,k$. Moreover, $n\in[0,k-1]$ and $a_{0k}=\frac{2k(k+2)}{k+1}$.
I have looked up some basic strategies for solving recurrence relations with variable coefficients, using generating functions and all of this seems fine. One can write $$ f(x,y)=\sum_{k=0}^{\infty}\sum_{n=0}^{k-1}a_{nk}x^ny^k $$ and with the regular methods of dividing by $x,y$ I can create $a_{n(k+1)}$ and $a_{(n+1)(k+1)}$ and with derivatives I can extract factors of $n$ and $k$. However, I am struggling to make sense of the term $a_{(n-k)k}$ in terms of generating functions. I know that the Cauchy product is often used to create such terms, but this only holds for infinite series, which this isn't for $n$. Does anyone know how to go about defining this in terms of a generating function or if there are better methods to try?