Related papers: On the Convergence of the Catalan Generating Funct…
Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) =…
A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a…
We introduce the super Patalan numbers, a generalization of the super Catalan numbers in the sense of Gessel, and prove a number of properties analagous to those of the super Catalan numbers. The super Patalan numbers generalize the super…
In this manuscript, we address the resolution of conformable Gegenbauer differential equations. We demonstrate that our solution aligns precisely with the results obtained through the power series approach. Furthermore, we delve into the…
Let G be the space of generating functions of a periodic infinite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.
The principal aim of this article is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions and derive their estimates developing the…
Correlation functions in a dynamic quartic matrix model are obtained from the two-point function through a recurrence relation. This paper gives the explicit solution of the recurrence by mapping it bijectively to a two-fold nested…
We present a higher genus generalization of $bc$-Motzkin numbers, which are themselves a generalization of Catalan numbers, and we derive a recursive formula which can be used to calculate them. Further, we show that this leads to a…
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the…
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on deriving recurrences. We show how to use…
We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the…
We study a recursion that generates real sequences depending on a parameter $x$. Given a negative $x$ the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a…
By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we…
Via a coupling argument, it is proved that the solution to a renewal equation has a power law decay rate in the case of a spread out interarrival distribution. By the regenerative property, the convergence in distribution for the recurrence…
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde…
The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…