Let $l_{an,n}$ be the least common multiple for $an+1, an+2, \dots, an+n$
I have found that for all $a$ and $n$ (details can be found here):
$$\frac{(an+n)!}{(an!)l_{an,n}} \le (n-1)!$$
This suggests to me that the following is true:
$$\frac{l_{an,n}}{n} \ge {{an+n}\choose{n}}$$
Here is my argument:
(1) $l_{an,n}[(n-1)!] \ge \dfrac{(an+n)!}{an!}$
(2) $\dfrac{l_{an,n}[(n-1)!]}{n!} \ge {{an+n}\choose{n}}$
(3) $\dfrac{l_{an,n}}{n} \ge {{an+n}\choose{n}}$