Let $n,p,q$ be non-negative integers. Then $$ \sum_{k=0}^n{2k+2p\choose k+p,k,p}{2(n-k)+2q\choose n-k+q,n-k,q}=4^n{2p\choose p}{2q\choose q}{n+p+q\choose n}.\quad\quad(\heartsuit) $$ In particular, for $p=q=0$ $(\heartsuit)$ reads as $$ \sum_{k=0}^n{2k\choose k}{2(n-k)\choose n-k}=4^n.\quad\quad\quad(\clubsuit) $$ There are known nice combinatorial (essentially bijective) proofs of $(\clubsuit)$, I wonder whether there is anything equally nice working for $(\heartsuit)$.

For what it worth, $(\heartsuit)$ is just Chu--Vandermonde identity $\sum {x\choose k}{y\choose n-k}={x+y\choose n}$ for $x=-p-1/2$, $y=-q-1/2$.

Let $n$, $p$, $q$ be non-negative integers. Then $$ \sum_{k=0}^n{2k+2p\choose k+p,k,p}{2(n-k)+2q\choose n-k+q,n-k,q}=4^n{2p\choose p}{2q\choose q}{n+p+q\choose n}.\tag{$\heartsuit$}\label{heart} $$ In particular, for $p=q=0$ \eqref{heart} reads as $$ \sum_{k=0}^n{2k\choose k}{2(n-k)\choose n-k}=4^n.\tag{$\clubsuit$}\label{club} $$ There are known nice combinatorial (essentially bijective) proofs of \eqref{club}, I wonder whether there is anything equally nice working for \eqref{heart}.

For what it's worth, \eqref{heart} is just Chu–Vandermonde identity $\sum {x\choose k}{y\choose n-k}={x+y\choose n}$ for $x=-p-1/2$, $y=-q-1/2$.