I have the following recurrence relation:
$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1}\right)$
This definition must be used under the condition $0<k\leq n^2$. The other base cases are the following:
- $b(n,k) = \begin{cases} 0 & \text{if } k\leq 0 \text{ or } k>n^2 \\ 0 & \text{if } n<1 \\ n^2 & \text{if } k=1 \end{cases}$
How can I check if a closed-form solution exists for the above recurrence? Is there any way to simplify the summation?
