Let $n,p,k$ be non-negative integers ($p$ prime, but not important here). I wondered if there is a more closed-form (i.e., simpler) formula for
$$A(p,n,k)=\frac{(pn)!}{(p!)^k(p(n-k))!}$$
and more precisely for
$$A(p,n)=\sum_{k=1}^n A(p,n,k).$$
I know that for $k=1$, $A(p,n,k)=\binom{pn}{p}$, but that's about it what I can figure out.