Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

I suspect some form of CLT for large $n, r$, which may possibly be proved by adopting Laplace's method of approximating the integrand by an appropriate gaussian kernel.

Interestingly enough, we can prove that:

$$ \int_{-\infty}^{\infty} \frac{\binom{n}{p}\binom{n-p}{r-2p}2^{r-2p}}{\binom{2n}{r}}\,\mathrm{d}p = 1. $$

To show this, we may apply the Legendre duplication formula to write

$$ \frac{\binom{n}{p}\binom{n-p}{r-2p}2^{r-2p}}{\binom{2n}{r}} = \frac{n!}{\binom{2n}{r}} \frac{\sqrt{\pi}}{\Gamma(1+p)\Gamma(n-r+1+p)\Gamma(\frac{r}{2}+\frac{1}{2}-p)\Gamma(\frac{r}{2}+1-p)}$$

and then apply the Ramanujan's beta integral (see the formula (5.3.14) of DLMF: 5.13).