TITLE: How to prove that $\binom{n}{1}\binom{n}{2}^2\binom{n}{3}^3\cdots \binom{n}{n}^n \leq \left(\frac{2^n}{n+1}\right)^{\binom{n+1}{2}}$?
QUESTION [5 upvotes]: How can we prove that $$\binom{n}{1}\binom{n}{2}^2\binom{n}{3}^3\cdots \binom{n}{n}^n \leq \left(\frac{2^n}{n+1}\right)^{\binom{n+1}{2}}$$

$\bf{My\; Try::}$ Using $\bf{A.M\geq G.M\;,}$ we get
$$\binom{n}{0}+\binom{n}{1}+\binom{n}{2} + \cdots+\binom{n}{n}\geq (n+1)\cdot \left[\binom{n}{0}\cdot \binom{n}{1}\cdot \binom{n}{2}  \cdots \binom{n}{n}\right]^{\frac{1}{n+1}}$$
So $$2^n\geq (n+1)\left[\binom{n}{1}\cdot \binom{n}{2}\cdots \binom{n}{n}\right]^{\frac{1}{n+1}}$$
How can I solve it after that? Help me.
Thanks.

REPLY [2 votes]: You have the right approach, just need a small detour. 
Hint  Start with:
$$\sum_k k \binom{n}{k} = n 2^{n-1} \tag{why?}$$
Now apply AM-GM and watch your exponentiation.