# Tagged Questions

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### Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Does anybody know how I can find a closed form for the expression $$\prod_{k=1}^n \binom{k^2+2k}{k^2}?$$ There are many ways to handle such things, but with sum instead of product. Here, I have no ...
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### Closed form of $n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$

$n$ is given, and it takes part in the following formula. $$n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$$ Is there a nicer way for expressing it? Without the summation sign?
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### Closed form expression for unusual sum of binomial coefficients

How do I get a closed form expression for $\sum_{i=c}^{n} i\binom{i}{c}$? Note that the index ranges over the upper values of the binomial, not the lower. I know computer algebra systems can give me ...
Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ... 3answers 221 views ### Is there a closed form for the sum$\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$? I am interested in finding a closed form for the sum$\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here? Thank you. 1answer 110 views ### Recursive and closed form solution for choosing$n$pairs/triplets.. of$kn$elements. I stumbled apon an interesting question: How many ways are there to arrenge$kn$elements into$n$sets,$k$elements each? There should be a recursive and closed form solution for$g_k(n)$. For ... 3answers 202 views ### Calculate$\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $Is any formula which can help me to calculate directly the following sum : $$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? }$$ Or can you help me for an approximation? Thanks :) 2answers 121 views ### Calculation of binomial sum$\displaystyle \sum_{r=1}^{n}r.\binom{n}{r}x^r.(1-x)^{n-r} = \;\;?$[closed] How can I calculate $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$ 0answers 197 views ### Double sum with binomial coefficients Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be ... 2answers 237 views ### Is there a closed form expression for the first half of the Binomial series? I'm looking for a closed form expression for the sum$P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$, where$n$is a given positive integer and$k$runs over nonnegative integers between$0$and ... 1answer 309 views ### No closed form for the partial sum of${n\choose k}$for$k \le K$? In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the ... 2answers 442 views ### modified$\sum{k{n \choose k}}\$ closed form expression
There is probably something stupidly simple I'm missing, but I'm trying to find a closed form for: $$2\sum_{k=1}^{(n-1)/2} k \, {n \choose k} \hspace{1cm} (n\textrm{ is odd})$$ Anyone know how to ...