<s>[INST] A problem of AM>GM 
If $n$ is a positive integer with $n> 1$, prove that
  $$2^{n(n+1)}>(n+1)^{(n+1)}\cdot\left(\frac{n}{1}\right)^n\cdot\left(\frac{n-1}{2}\right)^{(n-1)}\cdots\left(\frac{2}{n-1}\right)^2\cdot\frac{1}{n}$$

For solving it I have considered the numbers $\displaystyle{n+1,\,n,\,\frac{n-1}{2},\,\ldots,\frac{2}{n-1},\,\frac{1}{n}}$ with associated weights $n+1,n,n-1,\ldots,2,1$ and applied AM>GM
 $$\left[{(n+1)^2+n^2+\frac{(n+1)^2}{2}+\ldots+\frac{2^2}{(n-1)}+\frac{1}{n}}\over{\frac{(n+1)\cdot(n+2)}{2}}\right]^{\tfrac{(n+1)\cdot(n+2)}{2}}$$
but cannot solve it please help
 [/INST] Hint:
$$\left(\frac{n}{1}\right)^n\cdot\left(\frac{n-1}{2}\right)^{(n-1)}\cdots\left(\frac{2}{n-1}\right)^2\cdot\frac{1}{n} = \binom{n}{0} \binom{n}{1} \cdots \binom{n}{n}$$
where $\binom{n}{k}$ is the usual binomial coefficient.
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