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Thread: An Equality of Binomial Sums

  1. August 11th 2010, 07:33 AM #1
    Vandermonde
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    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}
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  2. August 13th 2010, 01:53 PM #2
    awkward
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    Quote Originally Posted by Vandermonde View Post
    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.
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  3. August 16th 2010, 12:25 AM #3
    Vlasev
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    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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