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

Question

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.

Did you not find the answers to your previous question helpful? — Aryabhata, Feb 24 '12 at 07:38
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 08:37
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

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

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.

1 Answers1