Prove that $2^{n(n+1)}>(n+1)^{n+1}\left(\frac{n}{1}\right)^n\left(\frac{n-1}{2}\right)^{n-1}\cdots \left(\frac{2}{n-1}\right)^{2}\frac{1}{n}$

Notice that $$ RHS = (n+1)^{n+1} \cdot \left(\frac{n}{1}\right) \cdot \left(\frac{n}{1}\cdot \frac{n-1}{2}\right) \cdots \left(\frac{n}{1}\cdot \frac{n-1}{2} \cdots \frac{1}{n}\right) = \prod_{k=1}^n \binom{n}{k} = \prod_{k=0}^n \binom{n}{k}. $$ Then, by AM-GM, $$ RHS = (n+1)^{n+1} \cdot \prod_{k=0}^n \binom{n}{k} \le (n+1)^{n+1} \cdot \left( \frac{\binom{n}{0}+\binom{n}{1}+\ldots+\binom{n}{n}}{n+1}\right)^{n+1} = LHS. $$ Equality holds only if $\binom{n}0=\ldots=\binom{n}{m}$; i.e. for $n=1$.