Identity for sum of squares of reciprocals of binomial numbers

Question

I am looking for identities for $$\sum_{k=0}^n \left(\frac {k! (n-k)!} {n!}\right)^2.$$ For reciprocal of power one there are several identities.

Edit:

Also see Sury et al paper formula 9.

Answer 1

The following identity is stated in Riordan Array Proofs of Identities in Gould’s Book by R. Sprugnoli in (5.2). It is attributed to Tor B. Staver.

\begin{align*} \sum_{k=0}^n\frac{1}{\binom{n}{k}^2}=\frac{3(n+1)^2}{2n+3}\cdot\frac{1}{\binom{2n+2}{n+1}}\sum_{k=1}^{n+1}\frac{1}{k}\binom{2k}{k} \end{align*}