Sum of square binomial coefficients [duplicate]
$$(1+x)^n(x+1)^n=(1+x)^{2n}$$
$$\left(\sum_{0\le r\le n}\binom nr x^r \right)\left(\sum_{0\le r\le n}\binom nr x^{n-r}\right)=\sum_{0\le r\le 2n}\binom {2n}rx^r$$
Compare the coefficients of $x^n$
This is an immediate consequence of Vandermonde's identity. For $m,n,r\in \mathbb{N}_0$,
$$ {m+n \choose r} = \sum_{k=0}^r {m\choose k}{n\choose r-k}$$
Now set $m=r=n$, and replace ${n\choose n-k}$ with ${n\choose k}$.