Sum over binomial coefficients I want to find out if the following sum has a closed form:
$$\sum_{j=0}^N{j+k-1\choose k-1}{N-j+k-1\choose k-1}$$
I tried to rewrite the sum such that I could use the following form of the Vandermonde identity:
$$\sum_{m=0}^n{m\choose j}{n-m\choose k-j}={n+1\choose k+1}$$
but I couldn't arrive at any result.

Hint: You are doing good! Using what you put there you will need $n-(j+k-1)=N-j+k-1$ implies $n=N+2k-2$ and you also have that $2k-2-(k-1)=k-1$ So take the $k$ in the second equation to be $2k-2.$
