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Math Help - An Equality of Binomial Sums

  1. #1
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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. #2
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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. #3
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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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