All Questions
30
questions
3
votes
0
answers
162
views
Closed form for $\sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$?
I've found this sum:
$$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$
The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can ...
- 1,940
0
votes
1
answer
56
views
Simplify $ \sum\limits_{i=0}^l{n\choose i}{m\choose l-i}$
Is there a nice way to simplify $ \sum\limits_{i=0}^l{n\choose i}{m\choose l-i}$? I tried to tinker a bit with telescope sums but it did not get me nowhere...
- 4,564
4
votes
1
answer
123
views
Asymptotic formula/closed form of $\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1) }{n+r+1}$
I need an asymptotic formula/closed form for the sum $$\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1) }{n+r+1}$$ where $n\in\mathbb{N}$
Denote $$S_n=\sum_{r=0}^{n}\frac{(-1)^r\binom{n}{r}\log(n+r+1)...
- 910
1
vote
2
answers
70
views
Find $\lim\limits_{n\rightarrow\infty}\sqrt[n]{\sum_{k=0}^{n} \frac{(-1)^k}{k+1}\cdot2^{n-k}\cdot\binom{n }{k}}$
Find
$$\lim_{n\rightarrow\infty}\sqrt[n]{\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1}\cdot2^{n-k}\cdot\binom{n }{k}}$$
My attemt: By the binomia theorem, we have
$$(1-x)^n={\sum_{k=0}^{n} \binom{n }{k} (...
- 3,586
6
votes
2
answers
220
views
Finding a closed form for $\sum_{k=1}^\infty\sum_{n=k}^\infty\left(\frac{(-1)^k}{k^3\binom{n+k}{k}\binom{n}{k}}(\frac1{n^2}-\frac1{(n+1)^2})\right)$
Consider the sum $$\sum_{n=k}^{N} \frac{1}{\binom{n+k}{k}\binom{n}{k}}\left(\frac{1}{n^2}-\frac{1}{(n+1)^2}\right) $$
Using the above or otherwise I need a closed form for $$\sum_{k=1}^{\infty}\sum_{n=...
- 910
1
vote
2
answers
66
views
Closed form for $\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x)$
I request a closed form for the following sum $$S(n)=\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x) $$
I tried using De Moivre's theorem $$\cos(nx)+i\sin(nx)=(\cos (x)+i\sin(...
user1103841
3
votes
1
answer
75
views
Asymptotic formula of $\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}{n\choose r}$
I need to find the asymptotic equivalence of the sum $$\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}
{n\choose r} $$ where ${n\choose r}$ is the binomial coefficient.
We have the binomial identity $$(1-x)^...
user1094088
2
votes
3
answers
162
views
Evaluating $\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I want to find the closed form of:
$\displaystyle \tag*{}\sum \limits_{k=1}^{\infty} \frac{\binom{4k}{2k}}{k^2 16^k}$
I tried to use the taylor expansion of $\frac{1}{\sqrt{1-x}}$ and $\frac{1}{\sqrt{...
1
vote
0
answers
36
views
How to bound $\sum_{ j=1}^n j^{n-j}?$ [duplicate]
I came across this sum in some research I'm doing. I need to bound
$$S_n=\sum_{j=1}^{n}j^{n-j}.$$
One bound I've managed is obtained by doing the following: Divide by n! and observe
$S_n/n!=\sum_{j=1}...
0
votes
1
answer
119
views
Simplification of expression (related to multinomial theorem)
Is there a simplified way to write the following:
$$\sum_{r_1+r_2+\cdots+r_n=t,r_k\in\mathbb{N}}\frac{t!}{r_1!r_2!\cdots r_n!}\prod_{k=1}^nf(k)^{r_k}$$
This is very similar to the multinomial theorem ...
- 436
1
vote
1
answer
40
views
Binome-like sum with even integers
I am trying to compute the following sum
$$ \sum_{k=0}^n C^{2k}_{2n} \frac{(2k)!}{k! 2^k} x^{2n - 2k} y^{2k}. $$
I don't have many ideas, I tried to play a little with the factorials, but it did not ...
- 1,185
3
votes
3
answers
210
views
I cannot prove that $ \sum_{k=0}^n \sum_{i=k}^n {n \choose k} {n+1 \choose i+1} = 2^{2n} $
I 've tried to calculate the internal sum first with no success.
$$
\sum_{k=0}^n \Bigg( {n \choose k} \sum_{i=k}^n {n+1 \choose i+1} \Bigg) =
2^{2n}
$$
Thank you in advance
- 496
0
votes
4
answers
94
views
zero's for odd and $\pm 1$ for even. Or $\pm 1$ for odd and zero's for even
What is the formula of the real function $f$ that satisfies
\begin{equation}
\sum^{n}_{k=0}{f}=1+0+(-1)+0+1+0+(-1)+0+\cdots
\end{equation}
or
\begin{equation}
\sum^{n}_{k=0}{f}=0+1+0+(-1)+0+1+0+(-1)...
- 61
7
votes
4
answers
296
views
The sum: $\sum_{k=1}^{n}(-1)^{k-1}~ [(H_k)^2+ H_k^{(2)}]~ {n \choose k}=\frac{2}{n^2}$
This attractive identity that
$$\sum_{k=1}^{n}(-1)^{k-1}~ [(H_k)^2+ H_k^{(2)}]~ {n \choose k}=\frac{2}{n^2}~~~(*)$$
emerged while doing numerics at Mathematica with harmonic numbers, binomial ...
- 43.6k
1
vote
1
answer
71
views
Closed form solution for $\sum_{j=0}^{n} {n \choose j}^{2}x^{j}$?
As the title suggests, I'm interested to see if there is a closed form solution to
\begin{equation}\label{q}
\sum_{j=0}^{n} {n \choose j}^{2}x^{j}~.
\end{equation}
This can be seen as a ...
user382669