2
votes
2answers
68 views

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 ...
0
votes
1answer
91 views

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?
2
votes
2answers
310 views

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 ...
0
votes
2answers
91 views

How can I compute this sum of binomial

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.
0
votes
0answers
74 views

Close formula for triple sum binomial coefficient

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} ...
1
vote
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.
1
vote
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 ...
5
votes
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 :)
-2
votes
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 $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$
2
votes
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 ...
4
votes
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 ...
2
votes
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 ...
1
vote
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 ...