Closed form expression sum-product of binomials

Question

Is it possible to find a closed form expression for

$$\sum_{j=1}^a\sum_{i=1}^{b} {i+j-1\choose j} {i+j-1\choose i},$$ where $a \geq 1$, and $b \geq 1$ are integers.

I couldn't apply any type of Vandermonde formula, neither Mathematica provides me a closed form...

Answer 1

The following equation may help you.

Note that for any $k>0$ $$ {2 k \choose k+1} = \sum_{i=1}^k {k \choose i}{k \choose k+1-i} = \sum_{i=1}^k {k \choose i}{k \choose i-1} $$