confirm if ${2n \choose n}=2^{2n}$

Question

please verify if the following is true. I have tried but I could not get $2^n$

$$\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}=2^{2n}$$

Answer 1

A simple contre-example :

Take $ n=2 $, $ \binom{2n}{n}=\binom{4}{2}=6 $, and $ 2^{2n}=2^{4}=16 \cdot $

Thus "$ \left(\forall n\in\mathbb{N}\right),\ \binom{2n}{n}= 2^{2n} $" is a false predicate.