What is $\frac{(an)!}{n!}$? How can $\frac{(an)!}{n!}$ be expressed in terms of $a!$, $n!$, $a$, $n$ (and maybe Pochhammers)?

If you don't insist on being exact, you can use Stirling's approximation.  $$\frac{(an)!}{n!}\approx \frac {(an)^{an}e^n}{n^ne^{an}}\sqrt{a}=a^{an}n^{an-1}e^{-(an-1)}\sqrt a$$ which will be very close-within about a factor of $1+\frac 1{12n}$
