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Let $n$ be a natural number, $m\in [-n, n]$. Let $p=0,\ldots, \frac{n+m}{2}$.

Show, that for all $p$, $$ {n \choose \left[{\frac{n+m}{2}}\right]}\geq \frac{2^{n+1/2}}{\sqrt{n-p/2}}. $$

Thank you for your help.

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5  
What is this $N$ that suddenly pops up? –  Jonathan Christensen Jan 28 '13 at 22:31
    
@Alex, do you mean $\forall n,m>N$? –  user45099 Jan 28 '13 at 22:41
    
Sorry it suppose to be $n$ instead of $N$. –  Alex Jan 28 '13 at 23:01
    
If $m=n$ and $p={n+m\over 2}=n$, then the right hand side is infinite.... –  Byron Schmuland Jan 29 '13 at 1:02
    
Sorry, its one more typo-it should be $p/2$ under the square root. –  Alex Jan 29 '13 at 1:55

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