# Thread:

1. ## An Equality of Binomial Sums

If $n\ge 0$, show that

$\displaystyle \sum_{k=0}^{n}\binom{2n}{k}\binom{2n-2k}{n-k} = \sum_{k=0}^{n}\binom{2n+1}{k} = \sum_{k=0}^{2n}\binom{2n}{k}$

2. Originally Posted by Vandermonde
If $n\ge 0$, show that

$\displaystyle \sum_{k=0}^{n}\binom{2n}{k}\binom{2n-2k}{n-k} = \sum_{k=0}^{n}\binom{2n+1}{k} = \sum_{k=0}^{2n}\binom{2n}{k}$
This is false for n=2.

3. The first one seems to the the odd one out. The last two evaluate to the same closed form expression. I suppose there is a typo in the first sum.
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