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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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What is $^nC_2$? –  draks ... Feb 24 '12 at 7:17
    
Did you not find the answers to your previous question helpful? –  Aryabhata Feb 24 '12 at 7:38
    
yes i did. sorry about this one, had another expression which was posing a bit of a problem for me, and not this. Lets call this a copying error. thankyou anyway. –  Hemantika Feb 24 '12 at 8:37
    
@Hemantika: You can try posting another question. –  Aryabhata Feb 24 '12 at 10:30
    
Please don't have titles that consist of nothing but Markup/$\LaTeX$. Also, a big expression is not a very informative title. –  Arturo Magidin Feb 24 '12 at 16:33

1 Answer 1

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.

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