How does one solve the recurrence relation $a(n,k) = a(n-1,k) +\sum_{1 \leq i \leq n-2, 1 \leq j \leq k-1}a(i,j)a(n-1-i,k-j) +a(n-1,k-1)$?
With initial conditions $a(1,1) = 1$ and $a(n,k) = 0$ for all $\lbrace (n,k): n < k \text{ or } n \leq 0 \text{ or } k \leq 0 \rbrace$, I computed the first few rows of the triangle and guessed (It was of great help to notice $a(n,2) = n(n-1)/2$, OEIS confirmed the hunch for larger $k$) the solution to be $a(n,k) = \binom{n}{k-1} \binom{n-1}{k-1}\frac{1}{k}$. Given the guess, it is possible to verify the recursion holds.
I wanted to ask: is there a general way for solving this type of recurrence relation that does not involve 'guessing and working backwards'?