TITLE: Proving formula for product of first n odd numbers
QUESTION [16 upvotes]: I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$):
$$\prod_{i = 1}^{n} (2i - 1) =  \frac{(2n)!}{2^{n} n!}$$
How can I prove that it holds (or find a counter-example)?

REPLY [3 votes]: $$\Pi_{k=1}^n(2k-1) = \Pi_{k=1}^n(2k-1) \frac{\Pi_{k=1}^n(2k)}{\Pi_{k=1}^n(2k)} =\frac{\Pi_{k=1}^{2n}k}{2^n\Pi_{k=1}^n k} =  \frac{(2n)!}{2^n n!}$$