Deriving Equality with Binomials

[LuHell]

Hi,

I am reading a paper and I am trying to understand an equality which is given without proof:
$$\sum_{k=1}^s\binom{2s-k}{s}\frac{k}{2s-k}v^k(v-1)^{s-k}=v\sum_{k=0}^{s-1}\binom{2s}{k}\frac{s-k}{s}(v-1)^{k}$$
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

 

[fresh_42]

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.