Closed form expression for $\sum\limits_{k=0}^{n}{n+k \choose k}{n+1 \choose k+1}X^{n-k}Y^{k}$?

Question

I was trying to solve a recurrence equation and I ended up with this: $$ A \sum \limits_{k=0}^{n} \frac{{n+k \choose k}{n \choose k}}{k+1} X^{n-k} Y^{k} $$

Now, I'm just wondering is there any way to simplify: $$ \sum \limits_{k=0}^{n} {n+k \choose k} {n+1 \choose k+1} X^{n-k} Y^{k} $$ where $ n, X, Y \in \mathbb{N} $.