Currently, I am trying to prove the following two identities, which arose as a result of my other question in the Math StackExchange recently:
\begin{equation} \sum_{j=0}^b\binom{b}{j}^2\binom{n+j}{2b}=\binom{n}{b}^2 \end{equation}
\begin{equation} \sum_{j=0}^b(-1)^j\binom{b}{j}\binom{n+j}{2b} = \left(-1\right)^b \binom{n}{b} \end{equation}
In particular, the first identity appeared as a remark in Volume 4 of Henry Gould's Combinatorial Identities.
Out of curiosity, I am wondering if there is a combinatorial proof for any of these two identities?
EDIT: I have managed to prove the second identity, by considering the coefficient of $x^{n-b}$ in the expansion of $(1+x)^{-(b+1)}=(1+x)^b(1+x)^{-(2b+1)}$.
EDIT 2: I have also managed to find a proof for the first identity by using the Vandermonde identity. I would still be interested however, in a proof of the second identity using a more combinatorial approach.