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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.
Generating functions related to Catalan words and frequencies of digits are obtained using continued fractions. This is fast, elegant, and flexible. It follows the philosophy of Philippe Flajolet from 1980.
In this article, we propose an integral expression of the Catalan numbers, based on Malmst\'en's definite-integral representation of $\ln\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained expression is likely to…
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…
Easily computable lower and upper bounds are found for the sum of Catalan numbers. The lower bound is proven to be tighter than the upper bound, which previously was declared to be only an asymptotic. The average of these bounds is proven…
In this paper we determine the parity of some sequences which are related to Catalan numbers. Also we introduce a combinatorical object called, \Catalan tree", and discuss its properties.
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
The classical platonist/formalist dilemma in philosophy of mathematics can be expressed in lay terms as a deceptively naive question: is new mathematics discovered or invented? Using an example from my own mathematical life, I argue that…
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.…
In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem…
We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a…
Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for…
We present an integral expression of the Catalan numbers, based on F\'eaux' integral representation of $\log\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained formula may be the starting point of the derivation…
The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes…
We present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The method has two {\it…
We consider a simple theoretical model to investigate the impact of inheritances on the wealth distribution. Wealth is described as a finite resource, which remains constant over different generations and is divided equally among offspring.…
In this paper we derive inferential results for a new index of inequality, specifically defined for capturing significant changes observed both in the left and in the right tail of the income distributions. The latter shifts are an apparent…
We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.
In this paper, we present three methods to give the value of a classical integer in $\lambda\mu$-calculus. The first method is an external method and gives the value and the false part of a normal classical integer. The second method uses a…
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also…