# Is there a closed form for $1+ \sum_{m=1}^{2(n-1)} \prod_{k=1}^{m} \dfrac{2(n-1)-(k-1)}{( ^nC_2 -k)}$?

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

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

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

$$1 + \sum_{m=1}^{N-1} \frac{\binom{N}{m}}{\binom{r}{m}} = \frac{r+1}{r-N+1} - \frac{1}{\binom{r}{N}}$$
$$\frac{r+1}{r-N+1}$$
where $r$ and $N$ were defined above.