The binomial-coefficients tag has no wiki summary.

**28**

votes

**3**answers

2k views

### A limit involving binomial coefficients?

Experimentation suggests the limit
$$\lim_{n\rightarrow\infty} (-1)^n\sum_{k=1}^n(-1)^k{n\choose k}^{-1/k}=\frac{1}{2}\ .$$
Does somebody have an idea for (a start of) a proof?
Added: There seem to ...

**6**

votes

**0**answers

302 views

### Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol
$(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$.
The ...

**6**

votes

**3**answers

450 views

### Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients \frac{(2d)!}{(d!)^2} in terms of a generating function
$\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$
by the Taylor series of ...

**36**

votes

**2**answers

3k views

### Alternating sum of square roots of binomial coefficients

Let
$$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$
It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...

**6**

votes

**0**answers

455 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
...

**1**

vote

**1**answer

306 views

### Limit of an infinite series, related to (generalised Newton's) binomial expansion

How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$:
...

**6**

votes

**1**answer

603 views

### Sums of binomials with even coefficients

While looking for a closed form of a expression I worked myself to a formula that resembles the Vandermonde convolution, but is summed over even binomial coefficients only.
...

**0**

votes

**2**answers

894 views

### non negative integer solutions : Diophantine Equations [closed]

I want to know the exact number of non-negative integer solutions of a1x1+a2x2+...akxk = n ...
I know that it is the co-efficient of x^n in (1-x^a1)^-1 * (1-x^a2)^-1 * ... (1-x^ak)^-1 ...
but whats ...

**0**

votes

**1**answer

749 views

### How to calculate such sum of product of binomial coefficient?

..I wonder if the following formula can be calculated?
$
\sum_{k=0}^m {m \choose k} {2k \choose n}
$

**7**

votes

**3**answers

757 views

### Binomial coefficient in Andrews' partition book

First of all, i think MathOverflow is a very great community to discuss math, either basic or advanced, and i'm glad to participate here. It's my first post, so i'm sorry if i did anything wrong, and ...

**0**

votes

**0**answers

169 views

### $q$-binomials in terms of ordinary binomials

Is there a known identity that expresses the $q$-binomial coefficients (also known as Gaussian binomial coefficients) in terms of the (ordinary) binomial coefficients?

**4**

votes

**1**answer

487 views

### What is the degree of a symmetric boolean function?

(previous title " Zero sum of binomials coefficients - a stronger version ")
This is a stronger version of another question.
Is there an $N\in \mathbb N$ and a sequence of non-constant functions $ ...

**6**

votes

**2**answers

684 views

### Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians,
I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...

**8**

votes

**2**answers

537 views

### Zero sum of binomial coefficients

Is there a function $p:\mathbb N\to \{ 1,-1 \} $ and a fixed $N\in \mathbb N$ such that for every $n \geq N$ we get:
$\sum _{i=0} ^{n} p(i)\binom {n}{i}=0$
?
Obviously $p(i)=(-1)^i$ works for ...

**3**

votes

**1**answer

621 views

### Yet another sum involving binomial coefficients

Let $k,p$ be positive integers. Is there a closed form for the sums
$$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$
$$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$
...

**2**

votes

**1**answer

234 views

### Bounding a hypergeometric sum

Is there a general method for finding upper bounds of hypergeometric sums?
The sum in particular I am trying to bound is $\sum_{j=1}^{k}\binom{k-1}{j-1}\binom{n-k-1}{j-1}4^{k-j}$.
This is a ...

**0**

votes

**2**answers

631 views

### Cosine of a Partial Sum

Does anyone know of a closed formula for $cos(\displaystyle\sum_{n=0}^m a_{n})$? I've seen formulas for $cos(\displaystyle\sum_{n=0}^\infty a_{n})$ and $tan(\displaystyle\sum_{n=0}^m a_{n})$, but the ...

**6**

votes

**3**answers

799 views

### Multiplicative Convolution for Binomial Coefficients

I know Vandermonde's convolution for binomial coefficients:
$$\sum_{j=0...k} \binom{n}{j} \binom{m}{k-j} = \binom{n+m}{k}$$
Is there a similar multiplicative convolution? More precisely, is there a ...

**0**

votes

**1**answer

1k views

### The Hexagonal Property of Pascal's Triangle

Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:
the product of non-adjacent vertices is constant.
the greatest common ...

**8**

votes

**7**answers

4k views

### Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...

**6**

votes

**0**answers

649 views

### Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...

**6**

votes

**2**answers

532 views

### Asymptotic difference between a function and its “binomial average”

(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n ...

**6**

votes

**1**answer

745 views

### Divisibility of a binomial coefficient by $p^2$ — current status

While skimming the book Concrete Mathematics, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96)
Is ${2n \choose n}$ ...

**3**

votes

**2**answers

618 views

### How to restore the original formula from a binomial-like expansion?

I encountered with a recursive formula of the following kind:
$$A(0,x)=1$$
$$A(n,x)= \sum _{j=0}^{n-1} \binom{n-1}{j} A(n-j-1,x) \sum _{k=0}^{x-1} A(j,k)$$
The sum terms can be re-arranged so to ...

**9**

votes

**3**answers

1k views

### Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$
with initial condition $R_0^0 = 1$ and ...

**3**

votes

**2**answers

4k views

### Combinatorial proof of a recurrence for the Catalan numbers

I would like to ask whether there is a combinatorial proof of the following recurrence relation for Catalan numbers:
$$
C_{n+1}=\frac{4n+2}{n+2} C_n.
$$
Thanks!~

**2**

votes

**1**answer

612 views

### Partial sums of the Chu--Vandermonde identity

I am interested in finding a lower bound of the sum:
$$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right)
\left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$
when $d < k$ (and assuming both $n\geq k$, ...

**17**

votes

**3**answers

2k views

### Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello.
I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...

**1**

vote

**2**answers

2k views

### bound for binomial coefficients

How can one show $\displaystyle\frac{(m+n-1)!}{m ! (n-1)!}\leq \left[\frac{e (m+n-1)}{n}\right]^{n-1}$ ?

**28**

votes

**3**answers

2k views

### A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from ApĂ©ry's biography
(which some of the people have already enjoyed in
this answer).
During a mathematician's dinner in
Kingston, Canada, in 1979, the
conversation turned ...

**10**

votes

**4**answers

1k views

### Rational congruence of binomial coefficient matrices

Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on.
Say that two matrices $A$ and $B$ over the ...

**32**

votes

**4**answers

2k views

### Integer-valued factorial ratios

This historical question recalls
Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation
Chebyshev used the factorial ratio sequence
$$
...

**13**

votes

**1**answer

1k views

### Binomial supercongruences: is there any reason for them?

One of the recent questions, in fact
the answer
to it, reminded me about the binomial sequence
$$
a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,
\qquad n=0,1,2,\dots,
$$
of the Apéry ...

**9**

votes

**3**answers

2k views

### Showing a matrix is negative definite [formerly Showing a sum is always positive]

For each $d$, I have a matrix $M$ with values
$$
M_{ij} = \begin{cases}
\frac{4ij}{d} - \binom{2d}{d} & i \neq j & \\\\
\frac{4i^2}{d} - \binom{2d}{d} -
...

**11**

votes

**5**answers

844 views

### Asymptotics of a Bernoulli-number-like function

Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = ...

**2**

votes

**3**answers

503 views

### Variant of binomial coefficients

I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc.
First recall the following. If z is a ...