# Questions tagged [binomial-coefficients]

The binomial-coefficients tag has no usage guidance, but it has a tag wiki.

**-4**

votes

**0**answers

28 views

### BInomial theorem [on hold]

Obtain and simplify the first three terms in the expansion of (1- (x/2))^6 in ascending powers of x. Given that the coefficient of x and x^2 in the expression of (2 + ax + bx^2)(1- (x/2))^6 are -5 and ...

**0**

votes

**3**answers

171 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

281 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

100 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 $X\sim B(n,p)$.

**11**

votes

**3**answers

932 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

572 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

150 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

98 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

109 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

110 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

1k 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 ...

**6**

votes

**1**answer

282 views

### Closed form expression for a recursion relation with binomial coefficients

I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k?\end{pmatrix} T_{k}, \?\?\?\ T_0 = C \in \mathbb{N} $$
I would like to express it as a function of n, ...

**1**

vote

**0**answers

76 views

### Has this result about the number of permutations of a given cycle type (or centralizers) been proved?

I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...

**2**

votes

**1**answer

155 views

### Number of odd elements in a vanishing sum of binomial coefficients

Let $n$ be a positive integer, $k$ a non-negative integer and $N(n,k)$ be the number of odd elements among the numbers $\binom{n+k}{j}\binom{-n-k}{n-j}$, $0\le{j}\le{n}$, which sum to $0.$ It seems ...

**1**

vote

**0**answers

66 views

### Shuffling unordered partitions

Consider the following:
Let $\mathcal{A}$ be an unordered partition of $\{1,\dotsc,p\}$,
Let $\mathcal{B}$ be an unordered partition of $\{1,\dotsc,q\}$
Let $\mathcal{C}$ be an unordered partition of ...