# Questions tagged [binomial-coefficients]

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232
questions

**4**

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**3**answers

280 views

### Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers.
Let there be $n$ pairs of shoes in a box.
The the probability that from ...

**1**

vote

**2**answers

111 views

### Monotonicity of $M$-sequence

Consider the following definition in the second page of this article:
For any two integers $k,n\ge 1$, there is a unique way of writing
$$n=\binom{a_k}{k}+\binom{a_{k-1}}{k-1}+\dots+\binom{a_i}{i}...

**7**

votes

**1**answer

264 views

### Voyage into the golden screen (sequence defined by recurrence relation)

We start from A004718 named "The Danish composer Per N?rg?rd's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation"
$$a(2n) = -a(n), \qquad a(2n+1) = a(n) + ...

**8**

votes

**1**answer

308 views

### Reciprocal sum of binomials and divisibility by $3$

We all know that $\sum_{k=0}^n\binom{n}k$ is not divisible by $3$.
QUESTION. Is it true that the numerator of $a_n$ (in reduced form) is never divisible by $3$?
$$a_n=\sum_{k=0}^n\frac1{\binom{n}...

**5**

votes

**0**answers

84 views

### Q-analogue of an inequality

Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$.
It is not super-difficult to prove the inequality
$$
\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.
$$
This is actually quite a nice inequality that was ...

**0**

votes

**3**answers

190 views

### How to calculate$ \sum \limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}$?

How to calculate $$\sum\limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}.$$

**3**

votes

**3**answers

306 views

### How to calculate: $\sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}$

How to calculate:
$$\sum _{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$

**4**

votes

**3**answers

115 views

### A clean upper bound for the expectation of a function of a binomial random variable

I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.

**12**

votes

**3**answers

988 views

### Does the set $\{\binom x3+\binom y3+\binom z3:\ x,y,z\in\mathbb Z\}$ contain all integers?

The Gauss-Legendre theorem on sums of three squares states that
$$\{x^2+y^2+z^2:\ x,y,z\in\mathbb Z\}=\mathbb N\setminus\{4^k(8m+7):\ k,m\in\mathbb N\},$$ where $\mathbb N=\{0,1,2,\ldots\}$.
It is ...

**8**

votes

**1**answer

591 views

### A curious inequality concerning binomial coefficients

Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it.
Let $a_1,a_2,\ldots,a_k$ be non-negative integers such that $\sum_i ...

**2**

votes

**0**answers

153 views

### A binomial coefficient identity

i'm unable to prove the following : $\forall n$ integer $\geq 3$,
$ \displaystyle \displaystyle \sum_{s=1}^n \sum_{j=n-s+1}^n \displaystyle \frac{ (\binom n j )^2 \binom {n+j} n }{s-n+j} ( \...

**1**

vote

**1**answer

103 views

### Question about arithmetic binomial coefficient

i have a question about the following assertion:
let $n,j,u $ positive integer satisfying
$ n \geq 5,$ $ 1\leq j \leq n-1$,$ \; n+1 \leq u \leq n+j$
let $ d[n]:=\operatorname{lcm}[1,2,..,n]$ ...

**0**

votes

**2**answers

112 views

### Specific partial sum of even/odd binomial coefficients

I have a following sum:
$S_g=\sum_{k=0}^g k\binom{4g+2}{2k}$
I can transform it into a different sum
$S_g=(2g+1)\sum_{k=1}^g\binom{4g+1}{2k-1}$
What is the closed form or what is the method to ...

**0**

votes

**1**answer

118 views

### Simplify a double summation involving binomial coeficient

$$T(N,K)=\sum_{i=2}^{K}\sum_{j=2}^{i}(-1)^{i-j}\binom{i}{j}\frac{j^{N+1}-1}{j-1}$$
Is it possible to evaluate the sum for $K=10^7$ efficiently. If we manage to remove one of the sums, it will be ...

**14**

votes

**2**answers

2k views

### Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$

Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...