Related papers: An involution for a Catalan-tangent number identit…
We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$…
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and…
It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a…
In this article, we use the Touchard identity in order to obtain new integral representations for Catalan numbers. The main idea consists in combining the identity with a known integral representation and resorting to the binomial theorem.…
By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this…
The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger…
In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for…
We analyze a weighted convolution of Catalan numbers $$ \sum_{k=0}^{n} \binom{2k}{k}\binom{2(n-k)}{n-k} a^k = \sum_{k=0}^{n} (k+1)(n-k+1) C_k C_{n-k} a^k, $$ emphasizing its combinatorial, analytic, and probabilistic aspects. We derive a…
Hexaflexagons were popularized by the late Martin Gardner in his first Scientific American column in 1956. Oakley and Wisner showed that they can be represented abstractly by certain recursively defined permutations called pats, and deduced…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
We establish combinatorial interpretations of several identities for the Catalan and Fine numbers and, along the way, we present some new bijections of independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k} (n+1)choose(2k+1)…
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting…
Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We…
Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…
For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of…
Two new identities about Catalan numbers are treated with Zeilberger's algorithm and Watson's hypergeometric series evaluation.
In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are…
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
In this paper, we define four transformations on the classical Catalan triangle $\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}$ with $C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}$. The first three ones are based on the determinant and the forth is…