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A Equality with binomials

  1. Dec 9, 2016 #1
    Hi,

    I am reading a paper and I am trying to understand an equality which is given without proof:
    [tex]\sum_{k=1}^s\binom{2s-k}{s}\frac{k}{2s-k}v^k(v-1)^{s-k}=v\sum_{k=0}^{s-1}\binom{2s}{k}\frac{s-k}{s}(v-1)^{k} [/tex]
    Here, s>0, k and v are positive integers.
    The equality in question appears in Lemma 2.1 of
    http://web.williams.edu/Mathematics.../graphs/mckay_EigenvalueLargeRandomGraphs.pdf

    Would you be kind and give me some insights on how to derive this equality?

    Thank you,

    LH
     
  2. jcsd
  3. Dec 9, 2016 #2

    fresh_42

    Staff: Mentor

    I'm afraid you will have to expand the terms ##(v-1)^{n}## and some addition theorems on binomials. Perhaps an induction on ##s## can shorten the way.
     
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