# Questions tagged [binomial-coefficients]

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

308
questions

**3**

votes

**1**answer

346 views

### Conjecture on bernoulli numbers and binomial coefficients

Crossposted from
https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients
In playing around with some formulas, I have come up with the following ...

**0**

votes

**2**answers

188 views

### In search of a combinatorial proof for a multinomial sum

There is this sequence listed on OEIS - named Domb numbers. I'm curious about
QUESTION. Is there a direct combinatorial proof for the identity
$$\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}
=...

**2**

votes

**1**answer

169 views

### Evaluations of three series involving binomial coefficients

Question. How to prove the following three identities?
\begin{align}\sum_{k=1}^\infty\frac1{k(-2)^k\binom{2k}k}\left(\frac1{k+1}+\ldots+\frac1{2k}\right)=\frac{\log^22}3-\frac{\pi^2}{36},\tag{1}
\end{...

**6**

votes

**0**answers

154 views

### Looking for a combinatorial proof for an identity involving $q$-Catalan triangles

Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Following my earlier post on MO, one fine colleague asked me if there is a $q$-analogue of the identity formed by the so-called Shapiro's ...

**12**

votes

**5**answers

2k views

### Looking for a combinatorial proof for a Catalan identity

Let $C_n=\frac1{n+1}\binom{2n}n$ be the familiar Catalan numbers.
QUESTION. Is there a combinatorial or conceptual justification for this identity?
$$\sum_{k=1}^n\left[\frac{k}n\binom{2n}{n-k}\right]^...

**0**

votes

**0**answers

80 views

### Sum of binomials with power coefficients

I have a sum of the form
$$f(n) = \sum_{i=0}^{ \lfloor \log_2 n \rfloor} \sum_{j=0}^{i} \binom{n}{2^j}.$$
Is there a closed formula or an approximation for this expression? What if $n = 2^m$ is a ...

**0**

votes

**1**answer

97 views

### Maximum-average prefix of binomial coefficients

For $k\in\mathbb N$, let
$$
f(k)=\max_{j\in\{1,\dots,k\}}\left\{\frac{\sum_{i=0}^{j-1}\binom{k}{i}}{j} \right\}.
$$
Is there a way to express $f(k)$ explicitly?
If not, what is $\lim_{k\to\infty}f(k)$?...

**0**

votes

**0**answers

121 views

### Binomial coefficient in a binomial coefficient

I am doing some research in combinatorics, and I found that I have to consider the following binomial coefficient :
$$ \binom{\binom{i}{j}}{k} $$
(In fact, I have to take the product for fixed $i,k$ ...

**3**

votes

**2**answers

226 views

### Is there a combinatorial reason for variable-independence of this binomial-coefficient identity?

Consider the following identity
$$\sum_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$
It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ...

**1**

vote

**1**answer

112 views

### An ambitiouser binomial coefficients sum

I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...

**-1**

votes

**3**answers

190 views

### Binomial Coefficients sum [closed]

Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.

**6**

votes

**2**answers

309 views

### Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$

As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...

**6**

votes

**1**answer

340 views

### Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$

Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...

**3**

votes

**1**answer

226 views

### Sum of product of binomial coefficients

I would like to compute the following sum:
$$
\sum_{k=0, \, k =odd}^{\min\{2n, m\}} {2n \choose 2n-k}{2m-2n \choose m-k}
$$
So far I can prove that
$$
\sum_{k=0, \, k =odd}^m {2n \choose 2n-k}{2m-2n \...

**8**

votes

**1**answer

403 views

### What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?

A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik &...

**5**

votes

**1**answer

266 views

### Two remarkable weighted sums over binary words

This question builds off of the previous MO question Number of collinear ways to fill a grid.
Let $A(m,n)$ denote the set of binary words $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_{m+n-2})$ consisting ...

**7**

votes

**0**answers

202 views

### Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$

I have been trying to get a lower bound on the following alternating sum but without much success:
$$
\sum_{j=1}^T (-1)^j e^{-j^2} j^k .
$$
For small values of $k$, this is easy because the first term ...

**7**

votes

**0**answers

312 views

### Do generalizations of the identity $\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) = 1 $ exist?

On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \...

**2**

votes

**1**answer

141 views

### Do identities exist for the binomial series $\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $?

While examining the product of two upper triangular matrices, I've found that the $(m,n)$'th entry of the resulting matrix amounts to: $$\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $$ when $n \...

**1**

vote

**1**answer

147 views

### Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme)

In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$
$$R_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1_{x\geq y},$$
which is the Poisson distribution pdf. (This is ...

**1**

vote

**1**answer

134 views

### Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?

On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...

**3**

votes

**1**answer

195 views

### Combinatorial Summation $\frac{1}{n} \sum_{k=n+1}^{2n} (k-n)\binom{2n}{k}$ [closed]

For a particular problem, I reached until this point where eventually I have to prove this summation
$$
\frac{1}{n} \left ( \binom{2n}{n+1} + 2\binom{2n}{n+2} + 3\binom{2n}{n+3} + \dots + n\binom{2n}{...

**1**

vote

**0**answers

83 views

### Upper bound $\sum_{i=1}^m \sum_{j=1}^n p_{i,j}(1-p_{i,j})$

Let
$$p_{i,j} = \frac{\sum_{l=i}^{i+j-1} {l-1 \choose i-1} {m+n-l \choose m-i}}{{m+n \choose n}}$$
I am interested in approximating/upper bounding the sum
$$ \sum_{i=1}^m \sum_{j=1}^n p_{i,j}(1-p_{i,j}...

**10**

votes

**4**answers

663 views

### Are there any identities for alternating binomial sums of the form $\sum_{k=0}^{n} (-1)^{k}k^{p}{n \choose k} $?

In equations (20) - (25) of Mathworld's article on binomial sums, identities are given for sums of the form $$\sum_{k=0}^{n} k^{p}{n \choose k}, $$ with $p \in \mathbb{Z}_{\geq 0}$. I wonder whether ...

**1**

vote

**0**answers

129 views

### An explicit solution to the congruence $x^2\equiv 14(\frac 3p)-(\frac p3)-12\pmod {p}$?

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Then
$$14\left(\frac 3p\right)-\left(\frac p3\right)-12=\begin{cases}1&\text{if}\ p\equiv1\pmod{12},
\\-25&\text{if}\ ...

**2**

votes

**3**answers

397 views

### Show that the ratio of limits converges to the nearest Riemann zeta zero except when the ratio is a singularity

Let $h(s,n)$ be:
$$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$
and let $g(s,n)$ be:
$$g(s,n)=\lim_{c\...

**6**

votes

**0**answers

155 views

### Some conjectural congruences involving Domb numbers

The Domb numbers are given by
$$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$
Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....

**0**

votes

**0**answers

47 views

### Interlaced binomial expansion

For $a_1,a_2,\ldots,a_n < 1$ let
$$f(a_1,a_2,\ldots,a_n) = \sum_{k_1,k_2,...,k_n = 1}^\infty \frac{(k_1 + k_2)!}{k_1! k_2!} \frac{(k_2 + k_3)!}{k_2! k_3!}\ldots \frac{(k_{n-1} + k_{n})!}{k_{n-1}! ...

**5**

votes

**1**answer

144 views

### A $q$-analogue of a characterization of polynomials by binomial coefficients

Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely ...

**3**

votes

**2**answers

133 views

### Partial sums of signed binomial coefficients

I don't know if this is true or not but I want this to be true and so far I don't have any counterexample.
Let $i$ be odd. Do there exist coefficients $a_k \in \{0,1\}$ such that
$$\sum_{k=1}^{i-1} (-...

**2**

votes

**0**answers

234 views

### Bijective proof of a combinatorial identity: $\sum\limits_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m$

Identity
\begin{equation}
\sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1}
\end{equation}
was used in an answer here. As shown in that answer, (1) easily reduces to
\begin{...

**6**

votes

**0**answers

135 views

### An inequality related to the numbers of faces of polytopes with d+2 facets

I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below.
Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...

**0**

votes

**1**answer

88 views

### Solve equation involving binomial coefficient

I have a problem that leads to the following equation:
$${x \choose k} = N$$
For some unknown $x$ and known constants $k$ and $N$. Here all numbers are natural numbers. I can solve this analytically ...

**5**

votes

**1**answer

273 views

### Bounding the probability that two binomials are equal

Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.
Let $B_{n,p}$ denote the usual binomial random ...

**2**

votes

**1**answer

161 views

### A good estimate for a binomial sum

Are there good estimate for the sums
$$1.\quad\quad\quad\quad\quad\sum_{i=1}^k\frac{\binom{2k}{i}}{i!}$$
$$2.\quad\quad\quad\quad\quad\sum_{i=1}^k\frac{\binom{2k}{i}\binom{2k}{2k-i}}{i!(2k-i)!}=\sum_{...

**0**

votes

**2**answers

194 views

### Solving a recurrence relation involving binomial coefficients

This question originates from a graph neural network architecture (see 1) in which an edge-labelled graph $G=(V,E,\eta)$ of size $|V|=n$ with $\eta:E\to \mathbb{R}^{s_0}$ is represented as a "tensor''...

**4**

votes

**1**answer

87 views

### Combined identity perturbation

I found the interesting inequality when I study hypergraph 2-coloring
$$\sum_{i+j=k} \binom{r-1}{i}\binom{r-1}{j}(1-p)^i(1+p)^j \leq \binom{2r-2}{k}$$
$0\leq i, j < r$, $0\leq p \leq 1$. I want to ...

**4**

votes

**1**answer

129 views

### A binomial coefficient identity involving two parameters

In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$:
$$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...

**3**

votes

**1**answer

303 views

### How to prove this combinatorial identity?

If $n \in \mathbb N \setminus \{0\}$ and $x,y,z \in \mathbb R$ such that $x+y+z=n-1$, show that
$$\dfrac{(-4)^n}{\binom{2x}{n}}\sum_{r+s=n,r,s\in Z}\dfrac{\binom{y}{r}\binom{y-a}{r}\binom{z}{s}\binom{...

**1**

vote

**1**answer

132 views

### Does this series, related to the Hasse/Ser series for $\zeta(s)$, converge for all $s \in \mathbb{C}$?

I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though.
Numerical evidence suggests that:
$$\lim_{N \to +\infty} \sum_{n=1}^N\...

**5**

votes

**1**answer

115 views

### Alternating binomial sum asymptotics

Let
$$
S_k=\sum_{j=0}^{\alpha k}(-1)^j\binom{k/2}{j}\binom{\alpha k^2-j k}{k}
$$
where $\alpha\in(0,1/2)$ is a constant.
I'm interested in understanding the asymptotic behaviour of $S_k$.
It would ...

**1**

vote

**0**answers

99 views

### Bell polynomial with variables 1 and 0

Let $B_{n,k}(x_1,\cdots,x_{n-k+1})$ be the Bell polynomial.
If $x_1=\cdots=x_{n-k+1}=1$, we know that $B_{n,k}(x_1,\cdots,x_{n-k+1})=S(n,k)$, where $S(n,k)$ is the Stirling number of second kind.
...

**0**

votes

**1**answer

115 views

### Local behavior of the Vandermonde convolution

An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{...

**1**

vote

**0**answers

90 views

### Divisibility properties of linear combinations of binomial coefficients [closed]

Let $p$ be a prime and $a_0,\ldots,a_n\geq 0$ be integers. Define
$$
S(a_0,\ldots,a_n)=\sum_{k=0}^n a_k\binom{n}{k}.
$$
I am trying to find out how much we know about
$$
v_p(S(a_0,\ldots,a_n)),
$$
...

**6**

votes

**2**answers

431 views

### An upper bound for the G.C.D. of $\binom{a}{3}$ and $\binom{b}{3}$

I can't seem to find anything in the literature on how to estimate the g.c.d. of $\binom{a}{k}$ and $\binom{b}{k}$. In particular, I would like to know why $\gcd(\binom{a}{3}, \binom{b}{3})\leq b \...

**2**

votes

**0**answers

106 views

### Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...

**12**

votes

**1**answer

394 views

### Yet another real-rooted polynomial

In this entry I asked for the real-rootedness of a polynomial, and two very interesting answers were given: one using Malo's theorem and the other a clever rewriting of the expression using Jacobi ...

**-1**

votes

**1**answer

118 views

### Evaluation of $\int \prod_{j=1}^u \frac{x+j}{j-x}~dx$

Let $I_{n,k} = \frac{(n+k)!}{k!(k-1)!(n-k)!}$. This is a sort of generalization of the Apéry's numbers, with $I_{n,n} =$ the $n$-th Apéry number. I am studying integrals of the form:
$$f_u(x)=\int \...

**21**

votes

**2**answers

1k views

### Real rootedness of a polynomial

Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...

**1**

vote

**1**answer

432 views

### Find closed-form expression to $f(n)$

For all $n \in \mathbb{N}$, set
$f(n)=
\begin{cases}
\min_{a\in\{\lceil n/2\rceil, \lceil n/2\rceil+1,\dots, n-1\}} \frac 1 4 \binom n a f(a) & \text{if $n\geq 4$}\\
1 & \text{otherwise}
\...