Divide something to make this factorial into "gapped" factorial? $(2n+1)!$ divide by what (in simple expression) can turn it into a "gapped" factorial: $1\times 3\times5\times\ldots\times(2n+1)$? In other words, how to turn $(2n+1)!$ to $1\times 3\times5\times\ldots\times(2n+1)$?

Divide out the unwanted factors $2,4,6,\ldots, 2n$. As those are even, we actually divide out $2$ $n$ times and then $1,2,3,\ldots,n$. So you have $$ \frac{(2n+1)!}{2^n\cdot n!}$$