I'm looking for ordinary generating functions for the above expression. If it doesn't exist, I'll be open to consider exponential generating functions as well.
Edit: I'm actually trying to simplify the following expression:
$$\sum_{p=1}^{f_{1}} {\left(\left(\sum_{i=0}^{p-1} {f_{2} \choose i}\right) {f_{1}\choose p} \left(\sum_{i=0}^{p}{f_{3}\choose i}\right)\right)}$$
Here, $f_1, f_2$ and $f_3$ are known beforehand.