Is there a combinatorial interpretation for the following sum?

Let $a(n,t)= \sum_{k=0}^n \binom{n}{k}^2t^k.$

Using generating functions it is easy to show that

$$\sum_{k=0}^n a(k,t) a(n-k,t)=1/2 \sum_{k=0}^n \binom{2n+2}{2k+1}t^k.$$

Is there also a combinatorial proof of this identity?

For $t=1$ this reduces to $\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$, for which various combinatorial proofs are known.