I'm trying to prove following term is integer for all $m,n \in \mathbb{N} :$
$$\frac{2^{m+n-1}\prod\limits_{k=1}^m(4k-3)\prod\limits_{l=1}^n(4l-1)}{(m+n)!}$$
I checked that if $m=n$ then this term is an integer, but I don't have any idea how to prove this.
I am trying to input many integers and still it is an integer.
Thanks for any hints.