<s>[INST] Proving that $\binom{2n}{n} > n2^n , \forall n \ge 4 $ I'm trying to prove the following statement: $\binom{2n}{n} > n2^n ,  \forall n \ge 4 $
This is my attempt at an inductive proof:
Let $P(n)$ be the following proposition: "$\binom{2n}{n} > n2^n ,  \forall n \ge 4 $"
Base case:
$\binom{2*4}{4} = 70 > 4*2^4 = 64$ so $P(4)$ is true.
Inductive step: 
(I assume as an inductive hypothesis that P(n) is true and try to show that P(n+1) is true)
$\binom{2(n+1)}{n+1} = \binom{2n+2}{n+1} = \binom{2n+1}{n+1} + \binom{2n+1}{n}$
$= \binom{2n}{n} + \binom{2n}{n+1} + \binom{2n}{n} + \binom{2n}{n-1}$
$= 2 \binom{2n}{n} + \binom{2n}{n+1}  + \binom{2n}{n-1}$
$= 2 \binom{2n}{n} + \frac{(2n)!}{(n+1)!(2n-(n+1))!}  + \frac{(2n)!}{(n-1)!(2n-(n-1))!}$
$= 2 \binom{2n}{n} +2 \frac{(2n)!}{(n+1)!(n-1)!} $
By the inductive hypothesis we have:
$ 2 \binom{2n}{n} +2 \frac{(2n)!}{(n+1)!(n-1)!} > 2n2^n +2 \frac{(2n)!}{(n+1)!(n-1)!} = (2^{n+1}) n + 2 \frac{(2n)!}{(n+1)!(n-1)!} $
And here I'm stuck. 
If I could prove that $\frac{(2n)!}{(n+1)!(n-1)!} \ge 2^n ,  \forall n \ge 4 $ the last step would show that P(n+1) is true. But I haven't been able to prove that. Can anyone help me ?
 [/INST] $$\frac{(2n)!}{(n+1)!(n-1)!} =\frac n{n+1}\cdot{2n\choose n}\ge \frac{n^2}{n+1}2^n$$

A more direct proof for a better claim:
$2n\choose n$ is the largest of $2n+1$ summands in the expansion of $(1+1)^{2n}$, hence
$$ {2n\choose n}\ge\frac1{2n+1}\cdot 4^n.$$
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