相关论文: Paperfolding and Catalan numbers
We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod…
We establish the asymptotic density of the Catalan numbers modulo 3 and modulo powers of 2.
A coding method using binary sequences is presented for different computation problems related to Catalan numbers. This method proves in a very easy way the equivalence of these problems.
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 study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the…
We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fu\ss-Catalan numbers, and to subgroup counting functions…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
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…
In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation…
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…
In this paper, we study Catalan numbers which can be represented by the p-adic integral on Zp and we investigate some properties and formulae related to Catalan numbers and special numbers.
This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula…
The Pascal matrix, which is related to Pascal's triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…
By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan…
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.
We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence.
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…
This note presents some results about Hankel determinants of backwards shifted Catalan-like numbers and related sequences.
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…