The combinatorial-identities tag has no usage guidance.

**45**

votes

**1**answer

2k views

### A = B (but not quite); 3-d arrays with multiple recurrences

Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan)
1
1 3
2 10 15
6 40 105 105
24 196 700 1260 945
...

**29**

votes

**2**answers

2k views

### Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture?

This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most number theorists and ...

**25**

votes

**4**answers

2k views

### A mysterious Heisenberg algebra identity from Sylvester, 1867

I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...

**23**

votes

**3**answers

3k views

### Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$

I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$.
Some clues that might work (kindly provided by ...

**17**

votes

**2**answers

2k views

### What role does Cauchy's determinant identity play in combinatorics?

When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \...

**17**

votes

**3**answers

2k views

### A binomial sum is divisible by p^2

This is a question I have since longer time, but I have absolutely no idea how to proceed on it.
Let $p>3$ be a prime. Prove that $\displaystyle\sum\limits_{k=1}^{p-1}\frac{1}{k}\binom{2k}{k}\...

**11**

votes

**3**answers

2k views

### Gauss sum (with sign) through algebra

Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$.
Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for $...

**9**

votes

**5**answers

1k views

### A sum involving derivatives of Vandermonde

Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$.
I am intersted in the calculation of the following expression for fixed $k$:
$$\sum_i (x_i)^k (d/dx_i)^k V(x_1 , ...

**9**

votes

**2**answers

653 views

### An identity involving an infinite integral with a sinh in the denominator

I recently encountered the rather appealing looking integral, which appears in the theory of random matrices :
$$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...

**8**

votes

**2**answers

1k views

### Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.

Considering the success of a previous question involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this ...

**7**

votes

**3**answers

656 views

### Combinatorial identities

I have computational evidence that
$$\sum_{k=0}^n \binom{4n+1}{k} \cdot \binom{3n-k}{2n}= 2^{2n+1}\cdot \binom{2n-1}{n}$$
but I cannot prove it. I tried by induction, but it seems hard. Does anyone ...

**7**

votes

**1**answer

667 views

### (0,1)-matrix congruence: is it known?

[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...

**6**

votes

**4**answers

330 views

### Permanent identities for special classes of matrices

The permanent $P(M)$ of a matrix $M$ of size $n$ is defined to be:
$$
P(M) := \sum_{\sigma \in S_n}\prod_{i=1}^n M_{i\sigma(i)}
$$
If you have a matrix of the form
$$
M_{ij} := A_i + B_j
$$
where ...

**6**

votes

**3**answers

712 views

### Binomial Identity

I recently noted that
$$\sum_{k=0}^{n/2} \left(-\frac{1}{3}\right)^k\binom{n+k}{k}\binom{2n+1-k}{n+1+k}=3^n$$
Is this a known binomial identity? Any proof or reference?

**6**

votes

**2**answers

771 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 ...

**6**

votes

**1**answer

598 views

### Double sum involving binomial coefficients

I came across a sum of binomial coefficients while trying to solve a problem involving $SU(2)$ group integrals. I am not able to solve it, nor I found a similar identity in the literature. I would ...

**5**

votes

**1**answer

323 views

### Alternating sums of alternate Stirling numbers

Does anybody know of any identities or combinatorial interpretations for alternating sums of alternate Stirling numbers?
I am particularly interested in expressions of the form:
$$\pm\sum_{k}(-1)^k|...

**5**

votes

**1**answer

713 views

### Number of Permutations with k-inversions and with a single clamped value

This question is cross-posted from math.stackexchange because it might be too technical.
Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\sigma\in S_n$ is the ...

**5**

votes

**0**answers

271 views

### When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation
$$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$
for some ...

**4**

votes

**0**answers

292 views

### Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II

Let us use the notations of my previous question about Tamarkin's problem.
Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$.
An element $f\in \mathbb Z^{\mathbb Z}$ is said to be $\ell$-average-...

**3**

votes

**2**answers

246 views

### equality of expressions for sum

Assume I have a chain of real numbers, s.th.
$x_0 < y_0 < x_1<y_1<x_2<\dots <x_n<y_n$.
I'm trying to explicitely solve the expression
$$ \sum_{i=0}^n \frac{\prod_{j=0}^n(x_j-y_i)...

**3**

votes

**2**answers

580 views

### Open problems and known identities involving sums

As many people here, I know of a few identities involving expressions of the type $\sum_{i}\ f(i)$, with "arbitrarily complicated $f(\cdot)$", as well as closed formulas in some cases.
I also know ...

**3**

votes

**2**answers

148 views

### Identities involving sums of Catalan numbers

The $n$-th Catalan number is defined as $C_n:=\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n}\binom{2n}{n+1}$.
I have found the following two identities involving Catalan numbers, and my question is if ...

**3**

votes

**1**answer

192 views

### system of complex equations

I am working on a system of complex equations The question is the following:
Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that
$$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} |a_j|...

**3**

votes

**1**answer

421 views

### p-adic poly-Bernoulli numbers

We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method.
But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\...

**2**

votes

**1**answer

154 views

### Inverse formula for counting marginals

I am interested in a formula which relating two functions over a multiset.
I have a multiset $X$ of sets where each element in $X$ is a set $x \subseteq \{1,2,\ldots,m\}$. Now I have two ``count'' ...

**2**

votes

**1**answer

304 views

### Rational Binomial Identity

Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:
$$n-i-1=(d-1)\sum_{l=1}^{n-i-1}\frac{\binom{n-i-1}{l}}{\binom{n-i+d-3}...

**2**

votes

**1**answer

151 views

### Alternating sign binomial identity [closed]

I recently noticed that for a triple of integers $k \geq 2$, $k \geq m \geq t \geq 1$, the following identity seems to hold
$\sum_{j=0}^{m-t} (-1)^{m-t-j}{k \choose j}{m-1-j \choose t-1}={k-t \choose ...

**2**

votes

**1**answer

640 views

### A formula combining Euler $\phi$ and $\gcd$

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...

**2**

votes

**2**answers

802 views

### Proving generating functions equality

What do you use to prove the following equality (and possibly more general ones of the kind)?
\begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} z_2^{s+t} = \sum_{a,...

**2**

votes

**0**answers

85 views

### A combinatorial sum involving ratios of binomials [closed]

Can anyone suggest how to prove the following (for $k \le n$):
$$\sum \limits_{s=0}^N \frac{\binom{n}{k} \binom{N-n}{s-k} }{\binom{N}{s}} = \frac{N+1}{n+1}$$
I am assuming it to be true, and ...

**2**

votes

**0**answers

130 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**2**

votes

**0**answers

74 views

### Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer.
Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let
$Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let
$...

**1**

vote

**2**answers

658 views

### Converting a recursive definition to an explicit one

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?
I've listed out the first few terms:
for $x=0,1,2,3,4,5,6, 7$
we have $a_x ...

**1**

vote

**2**answers

770 views

### An identity involving a sum of binomial coefficients

I am moving through a classic paper (On Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my ...

**1**

vote

**1**answer

148 views

### Evaluation of sum of factorials

Is there an evaluation of this sum (possibly involving gamma functions)? $k$ and $n$ are natural numbers and $x$ is real with $0<x<1$.
$$ \sum_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-...

**1**

vote

**1**answer

623 views

### Identity of binomial series with factorial.

I'm looking for a simple identity for the formula:
$$
\sum_{n = 0}^{p} \binom{p}{n} \cdot n! \cdot x^n
$$
In words, I have $p$ "players" who can choose to play or not (every player is represented by ...

**1**

vote

**0**answers

312 views

### Transfinite Sums Related to a Sequence

Hello,
Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products ...

**0**

votes

**1**answer

354 views

### Finite sums with Binomial and Catalan inverses

In a recent failed-post about some partial sums with respect to the Central Binomial and Catalan number the formulas
$$\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3 B_{n+1}}+\frac{1}{3}$$
$$\sum_{...

**0**

votes

**1**answer

147 views

### Is it possible to write identity for $ \{x(y^2-z^2)-y\}.\{u(v^2-w^2)-v)\}=a(b^2-c^2)-b$? [closed]

I asked this question in "math.stackexchange" but I did not get any response, so I put it here, maybe someone can help.
Is it possible to write identity similar to the identity
$$
(x^2+y^2)(u^2+v^2)=...

**0**

votes

**1**answer

94 views

### positive expression

Let
$$a_{n,k}=\sum_{s_i \geq 1 \atop \sum_{i=1}^{n-k} s_i \leq n} \frac{2^{n}}{(2(n-\sum_{i=1}^{n-k} s_i)+1)!\prod_{i=1}^{n-k} (2s_i)! }$$
for $0 \leq k \leq n-1$. Prove for $1 \leq k \leq n-1$ that
$...