Let: $$I=\{1,\dots,n\}.$$
What is the value of: $$\sum_{J\subseteq I} {\binom{g+2|J|-1}{2|J|-1}}$$
where $g$ is a positive integer and $|J|$ represents the number of elements of $J$.
Possible first step:
The given expression equals: $$\sum_{d=0}^n {\binom{n}{d} \binom{g+2d-1}{2d-1}}$$