Questions tagged [combinatorial-identities]

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11 votes
1 answer
973 views

Is it true that $\sum_{k=1}^\infty\frac{\binom{2k}k^2}{k16^k}(H_{2k}-H_k)=\frac23\sum_{k=1}^\infty\frac{\binom{2k}k^2H_{2k}}{(2k+1)16^k}$?

On Jan. 27, 2012, I conjectured the identity $$\sum_{k=1}^\infty\frac{\binom{2k}k^2}{k16^k}(H_{2k}-H_k)=\frac23\sum_{k=1}^\infty\frac{\binom{2k}k^2H_{2k}}{(2k+1)16^k},\tag{$*$}$$ where $H_n$ denotes ...
23 votes
5 answers
2k views

Identity involving Pochhammer symbol

I came across the following identity in my research: $$ \sum_{m=0}^s \frac{(-1)^m (a+2m)}{m!(s-m)! (a+m)_{s+1}}=\delta_{s,0} $$ where $(a)_n= a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol. One can ...
4 votes
1 answer
203 views

Any conjectures about Jack Littlewood-Richardson coefficients when Schur LR > 1?

Stanley famously conjectured ("Some combinatorial properties of Jack symmetric functions" Adv. in Math. (77) 1989, doi:10.1016/0001-8708(89)90015-7, MR1014073, Zbl 0743.05072) that the Jack ...
2 votes
1 answer
667 views

Transfinite sums related to a sequence

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 of all $...
6 votes
2 answers
705 views

Recreation with Catalan

Consider the well-known sequence $C_k=\frac1{k+1}\binom{2k}k$ of Catalan numbers. I came across the below identity while working with certain generating functions. I thought it might be of interest to ...
13 votes
1 answer
420 views

Four new series for $\pi$ and related identities involving harmonic numbers

Recently, I discovered the following four new (conjectural) series for $\pi$: \begin{align}\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^k\binom{3k}k}{k(3k-1)(3k-2)\binom{2k}k\binom{4k}{2k}}&=\frac{3\pi}2,\...
8 votes
3 answers
582 views

An identity involving polylogarithms

Recall that $$\mathrm{Li}_2(x):=\sum_{n=1}^\infty\frac{x^n}{n^2}.$$ I have found the following identity: \begin{equation}\begin{aligned}&\mathrm{Li}_2\left(\frac{-1-\sqrt{-7}}4\right)+\mathrm{Li}...
3 votes
1 answer
415 views

Identities for Bernoulli numbers

I arrived at this formula by inductive reasoning, but I don’t know how to prove it. For any natural numbers $m$ and $k=0,1,2,\ldots, m-1$, $B_i$ - Bernoulli numbers we have: $$\sum_{i=0}^k (-1)^{k-i}\...
1 vote
1 answer
176 views

Curious identity involving the number of perfect matchings of the complete graph

Can you prove (preferably combinatorially) the following identity for the total number of perfect matchings of the complete graph $K_{2n}$, where the edges in the matching are ordered, i.e., $\binom{...
1 vote
1 answer
226 views

A vanishing sum and related $p$-adic congruences

Recently I had a curious discovery. Namely, I have made the following conjectures. Conjecture 1. We have the identity $$\sum_{k=0}^\infty\frac{(10k-1)\binom{3k}k\binom{6k}{3k}}{(2k+1)512^k}=0.\label{1}...
8 votes
0 answers
315 views

A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$

Recently, I found a (conjectural) new series for $\sqrt3\pi$: $$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$ The series converges fast ...
6 votes
1 answer
418 views

Three conjectural series for $\pi^2$ and related identities

Recently, I found the following three (conjectural) identities for $\pi^2$: $$\sum_{k=1}^\infty\frac{145k^2-104k+18}{k^3(2k-1)\binom{2k}k\binom{3k}k^2}=\frac{\pi^2}3,\tag{1}$$ $$\sum_{k=1}^\infty\frac{...
6 votes
0 answers
221 views

A curious series for $L(2,(\frac{-3}{\cdot}))$

Let $$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
6 votes
0 answers
282 views

A new series for $\sqrt3/\pi$?

Recently, I conjectured the following identity: $$\sum_{k=0}^\infty\frac{(66k^2+37k+4)\binom{2k}k\binom{3k}k\binom{4k}{2k}}{(2k+1)729^k}=\frac{27\sqrt3}{2\pi}.\tag{1}$$ This can be easily checked ...
3 votes
3 answers
715 views

Ordinary partitions vs partitions into odd parts

Let $\mathcal{P}(n)$ be the set of all unrestricted partitions of $n$ while $\mathcal{O}(n)$ stand for the set of all partitions of $n$ into odd parts. We adopt the power notation for partitions $\...
6 votes
1 answer
403 views

A summation involving fraction of binomial coefficients

I need to prove the following statement. Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $ ...
4 votes
0 answers
92 views

"Convolving" a general Catalan with classical Catalan

Consider what is sometimes known as generalized Catalan sequence $$\mathcal{{\color{red}C}}_{a,b}:=\frac{2b+1}{a+b+1}\binom{2a}{a+b}.$$ Observe that $\mathcal{{\color{red}C}}_{n,0}$ reduces to the ...
4 votes
0 answers
200 views

Extract this constant term

Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term. For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
3 votes
1 answer
212 views

Seeking for a combinatorial argument for partition identities

Given an integer partition $\lambda$, introduce the following quantities: \begin{align*} c(\lambda)&=\sum_{i\geq1}\left\lceil\frac{\lambda_i}2\right\rceil, \qquad c_o(\lambda)=\sum_{i\geq1}\left\...
10 votes
2 answers
2k views

Proving an identity about Catalan numbers

$$C_{n} = \sum_{i=1}^n (-1)^{i-1} \binom{n-i+1}{i} C_{n-i}$$ Are there any good combinatorial proofs or algebraic proofs of this?
0 votes
0 answers
225 views

Looking for a combinatorial proof of an identity

I've come up with an interesting combinatorial identity (thanks to P. Belmans who precomputed the numbers and pointed out to me that they correspond to OEIS A002697): $$ \sum_{i=0}^{n-1}\binom{n+1-i}{...
0 votes
0 answers
256 views

An alternating sum involving a product of binomial coefficients

I encountered the sum below, where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $d$ are some given positive constants. Does anyone have an idea how to simplify it? $$ \sum\limits_{k=1}^{d} \frac{(-1)^{k-1}k}...
7 votes
1 answer
271 views

A reference for a sum found in Gould's Combinatorial Identities book

On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it ...
2 votes
1 answer
199 views

A Vandermonde like determinant with exponentials

Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
10 votes
0 answers
490 views

New series for $\zeta(5)$ involving second-order harmonic numbers

In 1997 T. Amdeberhan and D. Zeilberger proved that $$\sum_{k=1}^\infty\frac{(-1)^k(205k^2-160k+32)}{k^5\binom{2k}k^5}=-2\zeta(3).\tag{1}$$ In 2008 J. Guillera obtained that $$\sum_{k=1}^\infty\frac{(...
9 votes
1 answer
387 views

Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers

The classical rational Ramanujan-type series for $1/\pi$ have the following four forms: \begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1} \\\sum_{k=0}^\...
2 votes
2 answers
268 views

Ask for a proof of an identity involving the product of two Bernoulli numbers

It is well known that the Bernoulli numbers $B_{k}$ for $k\in\{0,1,2,\dotsc\}$ can be generated by \begin{equation*} \frac{z}{\textrm{e}^z-1}=\sum_{k=0}^\infty B_k\frac{z^k}{k!}=1-\frac{z}2+\sum_{k=1}^...
4 votes
0 answers
214 views

Curious double sum identity

The following identity seems to hold for $a>1$ : $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{ \frac{a^m}{m}\left( \frac{a^m}{m}+ \frac{a^n}{n} \right) } = \frac{a^2}{2(a-1)^4}$$ I've tested ...
8 votes
1 answer
530 views

Trivial (?) product/series expansions for sine and cosine

In an old paper of Glaisher, I find the following formulas: $$\dfrac{\sin(\pi x)}{\pi x}=1-\dfrac{x^2}{1^2}-\dfrac{x^2(1^2-x^2)}{(1.2)^2}-\dfrac{x^2(1^2-x^2)(2^2-x^2)}{(1.2.3)^2}-\cdots$$ $$\cos(\pi x/...
5 votes
1 answer
300 views

A conjectural permanent identity

Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...
1 vote
1 answer
211 views

Gaussian at $q=\pm1$, log-concave polynomials, Catalan numbers

Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ with $[0]_q!:=1$ and the Gaussian polynomials $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,\cdot\,[n-k]_q!}$. Adopt the convention that $\binom{n}k_q=0$ whenever $k&...
12 votes
0 answers
497 views

$q$-analogue of the multinomial theorem?

The $q$-binomial theorem states that $$ \prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k. $$ This identity is a $q$-analogue of the binomial theorem $$ (1+t)^n = \sum_{k=0}^n \...
3 votes
2 answers
415 views

Combinatorial identity concerning integral matrices with prescribed row sums and column sums

How to prove the following identity? Let $r = (r_1, r_2, \ldots, r_d)$ and $c = (c_1, c_2, \ldots, c_d)$ be sequences of natural numbers such that $s = r_1 + r_2 + \cdots + r_d = c_1 + c_2 + \ldots + ...
2 votes
1 answer
197 views

$q$-binomial sum, slightly

Recall that $[n]_{q}!=\prod_{j=1}^n\frac{1-q^{j}}{1-q}$ and $\binom{n}k_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}$. Then the $q$-binomial theorem states $$\sum_{k=0}^n\binom{n}k_qq^{\binom{k}2}=\prod_{k=...
6 votes
1 answer
161 views

An identity for rational functions leading to equations for multiple polylogarithms

The following identity is not hard to prove: $$ \sum_{1\leq i_1<i_2<\ldots <i_{2n}\leq N} (-1)^{i_1+\ldots+i_{2n}}\frac{(1-x_{i_1})(1-x_{i_3})\ldots(1-x_{i_{2n-1}})}{(1-x_{i_2})(1-x_{i_4}) \...
3 votes
1 answer
361 views

Is this combinatorial identity known? (of interest for random matrix theory)

While playing around with random matrices and I arrived at a different formula for the mean of the limiting normal distribution for a spectral CLT for sample covariance matrices. More precisely I have ...
10 votes
1 answer
530 views

Identities involving derangements and roots of unity

For a positive integer $n$, a derangement of $\{1,\ldots,n\}$ is a permutation $\tau$ of $\{1,\ldots,n\}$ with $\tau(j)\not=j$ for all $j=1,\ldots,n$. For convenience, we let $D(n)$ denote the set of ...
2 votes
1 answer
485 views

Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture

Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then \begin{eqnarray} &&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
6 votes
1 answer
506 views

A novel identity connecting permanents to Bernoulli numbers

For a matrix $[a_{j,k}]_{1\le j,k\le n}$ over a field, its permanent is defined by $$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}:=\sum_{\pi\in S_n}\prod_{j=1}^n a_{j,\pi(j)}.$$ In a recent preprint of mine, ...
1 vote
1 answer
211 views

A conjectural identity involving infinite series

Recently I formulated the following curious conjecture based on my computation. Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{k}\binom{2k+1}{2j}(1-x)^jx^{k-...
-1 votes
1 answer
335 views

Could you please confirm or deny two identities involving weighted Stirling numbers of the second kind?

In the paper [1] below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by \begin{equation*}%...
3 votes
2 answers
428 views

Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$

Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-...
29 votes
4 answers
3k 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 ...
15 votes
2 answers
965 views

A rather curious identity on sums over triple binomial terms

While exploring the Baxter sequences from my earlier MO post, I obtained a rather curious identity (not listed on OEIS either). I usually try to employ the Wilf-Zeilberger (WZ) algorithm to justify ...
10 votes
2 answers
454 views

Identity involving a quadratic term inside the Pochhammer symbol

This identity came up in my research: $$ \sum_{m=1}^n m^2 \frac{(\frac{xy}n + m-1)_{2m-1} (n+m-1)_{2m-1}}{(x+m)_{2m+1} (y+m)_{2m+1}} = \frac{n^2}{(x^2-n^2) (y^2 - n^2)}. $$ Here $n$ is a fixed ...
5 votes
3 answers
659 views

How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?

With the aid of the simple identity \begin{equation*} \sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^{k}}=2^n \end{equation*} in Item (1.79) on page 35 of the monograph R. Sprugnoli, Riordan Array Proofs of ...
7 votes
1 answer
232 views

Reference for permanent integral identity

$\DeclareMathOperator\perm{perm}\DeclareMathOperator\diag{diag}$Using MacMahon's master theorem, the properties of complex gaussian integrals, and Cauchy's integral theorem one can show that the ...
4 votes
1 answer
142 views

$0,1$-matrices with $1$ in every row/column vs. all $0,1$-matrices

Chapter 2, Exercise 25 of R. Stanley's "Enumerative Combinatorics" Vol. 1 asserts that $$ \sum_{m,n \geq 0} \left(\sum_{t \geq 0} f_i(m,n)t^i\right)\frac{x^m}{m!}\frac{y^n}{n!} = e^{-x-y}\...
12 votes
3 answers
864 views

Set partitions and permanents

Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd. ...
5 votes
3 answers
749 views

Identity with Pochhammer and harmonic numbers

This came out of some work on the digamma function. Let $(x)_k=x(x+1)\cdots(x+k-1)$ denote the Pochhammer symbol. Then, Question. Can you prove/disprove this identity? $$\pmb{\frac{(\frac12)_j^...