TITLE: Proving $\prod_{j=1}^n \left(4-\frac2{j}\right)$ is an integer
QUESTION [21 upvotes]: How do I show that the product $$\biggl(4 - \frac21\biggr) \cdot \biggl(4 - \frac22\biggr) \cdot \biggl(4 - \frac23\biggr) \cdots \biggl(4 - \frac2{n}\biggr)$$ is an integer for any $n \in \mathbb{N}$.


Source: www.math.muni.cz/~bulik/vyuka/pen-20070711.pdf

REPLY [8 votes]: Using Pascal identity
$$\displaystyle\binom{2n}{n}=\left(4-\frac{2}{n}\right)\cdot\binom{2n-2}{n-1}$$
one gets for your product
$$\displaystyle\frac{\binom{2}{1}}{\binom{0}{0}}\cdot\frac{\binom{4}{2}}{\binom{2}{1}}\cdots\frac{\binom{2n-2}{n-1}}{\binom{2n-4}{n-2}}\cdot\frac{\binom{2n}{n}}{\binom{2n-2}{n-1}}=\binom{2n}{n}$$
and this is an integer as required.