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Can there be a closed form representation for the expression

$ 1+ \sum_{m=1}^{2(n-1)} \prod_{k=1}^{m} \dfrac{2(n-1)-(k-1)}{( ^nC_2 -k)} $

It would simplify working with some equations I have. The current form in its full expansion is way too cumbersome.

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    Please don't have titles that consist of nothing but Markup/$\LaTeX$. Also, a big expression is not a very informative title.2012-02-24

1 Answers 1

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Using the answer to your previous question.

Setting $N = 2(n-1)$ and $r = \binom{n}{2} - 1$, what we have is

$ 1 + \sum_{m=1}^{N} \frac{\binom{N}{m}}{\binom{r}{m}}$

In your previous question, we had shown that

$ 1 + \sum_{m=1}^{N-1} \frac{\binom{N}{m}}{\binom{r}{m}} = \frac{r+1}{r-N+1} - \frac{1}{\binom{r}{N}}$

Thus what you seek is:

$ \frac{r+1}{r-N+1}$

where $r$ and $N$ were defined above.