Tagged Questions
13
votes
3answers
412 views
Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$
How can we prove that
$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$
where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$.
This inequality is very interesting. I ...
11
votes
0answers
205 views
Binomial sum of $n$ terms in closed form
Can we calculate the given combinatorial sum in closed form?
$$ \frac{\binom{2}{0}}{1}+\frac{\binom{4}{1}}{2}+\frac{\binom{8}{2}}{3}+\frac{\binom{16}{3}}{4}+\cdots+\frac{\binom{2^n}{n-1}}{n}$$
10
votes
3answers
132 views
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
Prove
$$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$
I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its complement ...
10
votes
7answers
307 views
Summation simplification $\sum_{k=0}^{n} \binom{2n}{k}^2$
$\sum_{k=0}^{n} \binom{2n}{k}^2$
So i'm trying to simplify this one and I'm stuck in nowhere. Some kind of tip would be appreciated.
Thanks! :)
10
votes
6answers
110 views
A limit on binomial coefficients
Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.
What I can do is just use Stolz formula. But I could not proceed.
10
votes
2answers
176 views
Asymptotics of the sum of squares of binomial coefficients
We are trying to estimate the cardinality $K(n,p)$ of so-called Kuratowski monoid with $p$ positive and $n$ negative linearly ordered idempotent generators. In particular, we are interesting in the ...
8
votes
2answers
117 views
A sum with binomial coefficients
Show that $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^{n+2}=\frac{2^{n}n(n+2)!}{6}.$$
7
votes
1answer
181 views
Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$
I am having trouble proving that
$$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$
I know that
$$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty ...
6
votes
5answers
252 views
Simplify $\sum_{i=0}^n (i+1)\binom ni$
Simplifying this expression$$1\cdot\binom{n}{0}+ 2\cdot\binom{n}{1}+3\cdot\binom{n}{2}+ \cdots+(n+1)\cdot\binom{n}{n}= ?$$
$$\text{Hint: } \binom{n}{k}= \frac{n}{k}\cdot\binom{n-1}{k-1} $$
6
votes
1answer
127 views
Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?
Wolframalpha tells me that
$$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$
However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
6
votes
1answer
244 views
Vandermonde's-like identity
The Vandermonde's identity gives $$\sum_{k=0}^r \binom{m}{k}\binom{n}{r-k}=\binom{m+n}{r}.$$
Here is an example of Vandermonde's-like identity: For all $0 \le m \le n$,
$$\sum_{k=0}^{2m} ...
5
votes
3answers
157 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 :)
5
votes
2answers
57 views
Binomial probability with summation
Show that
$$\sum_{k=0}^{m} \frac{m!(n-k)!}{n!(m-k)!} = \frac{n+1}{n-m+1}$$
Attempt:
It becomes:
$$\sum_{k=0}^{m } \frac{\binom{m}{k}}{\binom{n}{k}}$$
Telescoping, pairing, binomial theorem don't ...
5
votes
2answers
108 views
Generalization Of The Binomial Theorem
Consider the sum
$$\sum_{k=0}^{n_0} {n \choose k} \cdot \alpha^k$$
where $\alpha \in \mathbb{R}$ arbritary, $n_0 < n$. So it looks like binomial theorem,
$$\sum_{k=0}^n {n \choose k} \cdot ...
5
votes
1answer
81 views
$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof
Is there a simple combinatorial proof for the following identity?
$$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$
where $m,j \geq 0$, $k \geq n \geq 0$.
