An Equality of Binomial Sums

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August 11th 2010, 07:33 AM
Vandermonde

An Equality of Binomial Sums

If $n\ge 0$ , show that

$\displaystyle \sum_{k=0}^{n}\binom{2n}{k}\binom{2n-2k}{n-k} = \sum_{k=0}^{n}\binom{2n+1}{k} = \sum_{k=0}^{2n}\binom{2n}{k}$
August 13th 2010, 01:53 PM
awkward

Quote:

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.
August 16th 2010, 12:25 AM
Vlasev

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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