Background
I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. This results is an expression for a sum involving square of a binomial coefficient
Problem I need to find a closed expression for $\sum_{k=0}^{\frac{l}{2}} \binom{\frac{l}{2}}{k}^2 p^{2k}$$$\sum_{k=0}^{l/2} \binom{l/2}{k}^2 p^{2k}$$
where $p$ is a function of $l$ and lies between 0 and 1.
So far I've found a closed expression for $\sum_{k=0}^{n}k^2 \binom{n}{k}^2$$$\sum_{k=0}^{n}k^2 \binom{n}{k}^2$$
Any suggestions are very much appreciated.
