All Questions
38
questions
1
vote
1
answer
129
views
Prove $\sum_{k=j}^{\lfloor n/2\rfloor}\frac1{4^k}\binom{n}{2k}\binom{k}{j}\binom{2k}{k}=\frac1{2^n}\binom{2n-2j}{n-j}\binom{n-j}j$
let $x>1$, $n\in \mathbb N$ and
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}(x+\sqrt{x^2-1}\cos{t})^ndt.$$
Prove that
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}\dfrac{1}{\left(x-\sqrt{x^2-1}\cos{t}\right)^{...
- 137
2
votes
2
answers
243
views
Is $p\geq 3$ a prime divisor of $\sum_{t=0}^{\frac{p-1}{m}-1}\sum_{s=0}^{\frac{p-1}{m}-1}\binom{tm}{sm}$ where $m$ divides $p-1~?$
Is it always true that $p\geq 3$ not a prime divisor of
\begin{equation*}
\sum_{t=0}^{\frac{p-1}{m}-1}\sum_{s=0}^{\frac{p-1}{m}-1}\binom{tm}{sm}
\end{equation*}
where $m$ divides $p-1~?$
The ...
- 1,366
3
votes
1
answer
331
views
Simplify $\sum_{j=0}^{m}\sum_{\ell=0}^{p}(-1)^{j+\ell}\binom{j+k}{j}\binom{2(k+m)-p}{2(k+j)}\binom{\ell}{m-j}\binom{2m+k}{p-\ell}$
Let $m\in\mathbb{Z}_{>0}$ and $p\in\mathbb{Z}$ with $0\leqslant p\leqslant 2m.$ Then how could we simplify the following double sum?
\begin{align*}
\sum_{j=0}^{m}\sum_{\ell=0}^{p}(-1)^{j+\ell}\...
- 1,366
2
votes
2
answers
94
views
How to prove that for any $k$ between $2$ and $n-2$ we have $\binom{n}{k} \ \ge \binom{n}2$
I am trying to show that for any $k$ from $2$ to $n-2$ we have $\binom{n}{k} \ \ge \binom{n}2$.
What I tried
I started from the fact that $\binom{n}{k} \le \binom{n}{k+1}$ if $k \le \frac{n-1}{2}$ and ...
- 39
4
votes
2
answers
234
views
Evaluating $\lim_{n\to\infty} \left(\prod_{r=0}^n \binom{n}{r}\right)^\frac{1}{n(n+1)}$
Problem statement:
If
$$ A = \lim_{n\to\infty} \left(\prod_{r=0}^n \binom{n}{r}\right)^\frac{1}{n(n+1)}, $$
Find $A^2$.
Solution:1)
\begin{align*}
\prod_{k=0}^{n} \binom{n}{k}
&= \prod_{k=1}^{n} \...
1
vote
1
answer
132
views
Finding a closed form for a limit of a sequence
Consider a sequence $$u_n=\sum_{r=1}^{n} \frac{{n\choose r} f^{(r)}(1)}{(r-1)!} $$ where $f$ is an infinitely differentiable real valued function on $\mathbb
{R}$.
Question: Given that the limit ...
user1063508
0
votes
1
answer
254
views
The lower bound about the binomial coefficient
From easy computation we can get
\begin{align*}
{n\choose{\ell}}=\frac{n}{\ell}\cdot\frac{n-1}{\ell-1}\cdots\frac{n-(\ell-1)}{1}\geq\frac{n^{\ell}}{\ell^{\ell}},
\end{align*}
where the last inequality ...
- 1,366
4
votes
1
answer
170
views
What is $\frac{\binom{n}{1}}{1} + \frac{\binom{n}{2}}{2} + \frac{\binom{n}{3}}{3} + \cdots + \frac{\binom{n}{n}}{n}$
Good Day
Today, I learnt that $$\frac{\binom{n}{0}}{1} + \frac{\binom{n}{1}}{2} + \frac{\binom{n}{2}}{3} + \cdots + \frac{\binom{n}{n}}{n + 1} = \frac{2 ^ {n + 1} - 1}{n + 1}$$
I changed it a little ...
- 1,858
3
votes
5
answers
242
views
Evaluating $ \sum_{r=0}^\infty { 2r \choose r} x^r $
I recently came across this question during a test
Evaluate the following $$ \sum_{r=0}^ \infty \frac {1\cdot 3\cdot 5\cdots(2r-1)}{r!}x^r $$
I converted it to the follwing
into $\sum_{r=0}^\infty {...
4
votes
2
answers
104
views
Prove that $\sum_{k=0}^n(-1)^k{n \choose k}\frac{1}{k+m+1}=\sum_{k=0}^m(-1)^k{m \choose k}\frac{1}{k+n+1}$.
I actually found this question in a calculus exercise, so I thought maybe it is an idea to convert an infinite sum to a Riemannian Integral.
But then, I realized that it was missing the $\lim_{n \...
4
votes
2
answers
258
views
Binomial identity involving central binomial coefficient
I came across this nice binomial identity
$${\sum}_{k=0}^{2n} \frac{(-1)^k {2n \choose k} {2k \choose k}}{n+k \choose k} = 1$$
This is equivalent to the hypergeometric function ${}_2 F_1(-2n, 1/2; n+...
- 178
2
votes
2
answers
184
views
Induction for binomial coefficients
I would like some help to prove the following equality :
$$\sum_{i=0}^n \binom{n}i^2=\binom{2n}n$$
I wanted to do a proof by induction :
$$\sum_{i=0}^{n+1} \binom{n+1}i^2=1+\sum_{i=1}^{n+1} \binom{n+1}...
- 1,952
4
votes
4
answers
337
views
Hint for Spivak's Calculus Problem 2-4-a
(note: this is very similar to a related question but as I'm trying to solve it without looking at the answer yet, I hope the gods may humor me anyways)
I'm self-learning math, and an answer to an /r/...
- 43
1
vote
2
answers
65
views
Simple permutations: approach and proof explanation for an exercise not very clear
Set $A = (a_1, a_2, a_3), B = (b_1, b_2, b_3)$ and $C = (c_1, c_2)$ are assigned.
How many are the ordered sequences formed by $5$ distinct elements containing $2$ elements of $A$, two elements of $B$,...
- 7,792
3
votes
2
answers
143
views
Source for similarity between Leibniz formula and binomial theorem?
To explain the similarity between the Leibniz formula and the binomial theorem, one answer by Tad uses paths to justify the similarity in the coefficients of the two formulae.
I understand the answer ...
user768754