Proof of an identity involving binomial coefficients

Question

I have found numerically that the following identity holds: \begin{equation} \sum_{n=0}^{\frac{t-x}{2}} n 2^{t-2n-x}\frac{\binom{t}{n+x}\binom{t-n-x}{t-2n-x}}{\binom{2t}{t+x}} = \frac{x^2+t^2-t}{2t-1}, \end{equation} where $n$, $t$, and $x$ are positive integers ($x \leq t$). To make it more visible, values of $n$ range from $0$ to $\frac{t-x}{2}$.

Any clue about how to prove it?

Thanks, Antonio

Answer 1

with the help of Maple i got the following result $\frac{4^{t-1} (t-x-1) \Gamma \left(t-\frac{1}{2}\right) \Gamma (x+2) \binom{t}{x+1}}{\sqrt{\pi } \binom{2 t}{t+x} \Gamma (t+x+1)}$