[Euler-Fermat]

$$\prod_{i = 1}^{n} (4-\frac{2}{i} )$$
$$= \prod_{i = 1}^{n} (\dfrac{4i-2}{i} )$$
$$=\prod_{i = 1}^{n} (\dfrac{2^n \bullet (1\bullet 3 \bullet 5 ....\bullet (2n-1))}{n!} )$$
$$=\prod_{i = 1}^{n} (\dfrac{2^n \bullet(2n-1)!}{n!} )$$
$$=\prod_{i = 1}^{n} (\dfrac{2^n\bullet n! \bullet(2n-1)!}{(n!)^2} )$$
$$=\binom{2n}{n} $$