Find $\sum_{n=0}^k \frac{C_n^k A^n B^{k-n} n!}{(\frac{n}{2})!}$.

Question

$$ \mbox{What does}\quad \sum_{n = 0}^{k}{k \choose n}A^{n}B^{k - n}\,{n! \over \left(n/2\right)!}\quad \mbox{converge to ?} $$

I know it converges (I have found an upper bound for it but I can't work out what it converges to?

I am using the definition: $\left(n/2\right)!=\Gamma\left(n/2 + 1\right)$.