I am attempting to simplify the following expression for integers $p\geq2$:
$$\sum_{k=0}^n \binom{kp}{\underbrace{k,\dots, k}_{p\text{ times}}}\binom{(n-k)p}{\underbrace{n-k, \dots, n-k}_{p\text{ times}}}$$
For the case $p=2$, I was able to use the idea from here using generating functions, which leads to $\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=4^n$.
I was hoping there would be a simple extension like $p^{pn}$ but it does not seem to be the case. I tried using the generalized Multinomial Theorem to re-create the proof of the case $p=2$ but to no avail. I do not necessarily need to worry to much about this value in the end since I can compute it numerically, but it would be nice to find a closed-form expression or at least bounds on it.
