I have the following two variable recurrence equation for integers $j,k$:
$f(j,k) = (k/j)f(j-1,k-1) - (3 + k/j)f(j-1,k+2)$
where $f(j,0) = (3^j - 1)/j + 3jf(j-1,2)$, $f(0,0) = 0$, $f(0,k) = a_k$.
I would like to express $f(n,0)$ in terms of the different $a_k$ values.
I tried a generating function approach, by considering the generating function $\phi(x,y) = \sum_j\sum_k f(j,k)x^jy^k$, but it gives rise to a nasty differential equation.
Any ideas on how to attack this recurrence equation? Maybe by just calculating it for a few $n$ values, observe a pattern and use induction? The latter is also quite tedious. Is there any neat approach to this?