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Consider $$ \prod_{j=0}^{i-1}\binom{n-2j}{2}\left[1+\sum_{k=0}^{N-1}{\left( \prod_{l=0}^{k} \dfrac{N-l}{M-l}\right)\left(1 + \sum_{m=k+1}^{N-1} \prod_{p=k+1}^m \dfrac{N-p}{M-p}\right)+1}\right] $$

where $N=2i(n-(i+1))$ and $M=\binom{n}{2}-i.$

Could this expression be simplified in some way?

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The ideas here might be applicable: math.stackexchange.com/questions/111803/… –  Aryabhata Mar 11 '12 at 20:58
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Why is this tagged graph-theory? If it has some origin in an expression evaluated over a graph, saying what that origin is might help someone answer the question. –  Peter Taylor Mar 12 '12 at 22:26
    
The sums look hypergeometric, perhaps the techniques in the book "A= B" by Wilf and Zeilberger (available on line), or look for Gosper or Zeilberger in your CAD are applicable. They will tell you if a closed form is available or not, and hand it to you. –  vonbrand Apr 6 at 15:06
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