Results 11 to 20 of about 1,655,826 (313)
Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation [PDF]
Journal of Mathematics, 2020 In this paper, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that are associated with the sequence of Catalan–Qi numbers and several sequences of series involving the Catalan–Qi numbers.
Wathek Chammam
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Journal of Combinatorial Theory, Series A, 1985 Let \(C_ n\) be the set of Catalan words: binary words with n zeros and n ones such that each initial segment has at least as many zeros and ones. For \(w\in C_ n\), let d(w) be the number of descents, a(w) be the sum over all descents of the number of zeros to the left, b(w) be the sum over all descents of the number of ones to the left.
J. Fürlinger, Josef Hofbauer
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Integral Representations of the Catalan Numbers and Their Applications
Mathematics, 2017 In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations ...
Bai-Ni Guo, Feng Qi, Guo Bai-Ni
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On the Catalan Numbers and Some of Their Identities [PDF]
Symmetry, 2019 The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers.
Wenpeng Zhang
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Congruences with q- generalized Catalan numbers and q-harmonic numbers [PDF]
Hacettepe Journal of Mathematics and Statistics, 2021 In this paper, we give some congruences related to q − generalized Catalan numbers, q − harmonic numbers and alternating q − harmonic numbers, using combinatorial identities and some known congruences.
N. Ömür, Z. Gür, S. Koparal
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Catalan-like Numbers and Determinants [PDF]
Journal of Combinatorial Theory, Series A, 1999 The Catalan numbers play a central role in enumerations. They can be defined by recursion but also via so-called Hankel matrices. The Motzkin numbers are defined by a very similar recursion, and Aigner also found a description using Hankel matrices. Both types of numbers are involved in several classical formulae with binomial coefficients (and with ...
Aigner, Martin
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Catalan pairs: A relational-theoretic approach to Catalan numbers [PDF]
Advances in Applied Mathematics, 2010 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Filippo Disanto +3 more
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Parametric Catalan numbers and Catalan triangles [PDF]
Linear Algebra and its Applications, 2013 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
He, Tian-Xiao
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Modular Fuss-Catalan numbers [PDF]
Discrete Mathematics, 2022 The modular Catalan numbers $C_{k,n}$, introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of $x_0 * x_1 * \dots *x_n$ where $*$ is a binary $k$-associative operation and $k$ is a positive integer. The classical notion of associativity is just 1-associativity, in which case $C_{1,n} = 1$ and the size of the unique class ...
Msapato, D
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Convolution identities involving the central binomial coefficients and Catalan numbers [PDF]
Transactions on Combinatorics, 2021 We generalize some convolution identities due to Witula and Qi et al. involving the central binomial coefficients and Catalan numbers. Our formula allows us to establish many new identities involving these important quantities, and recovers some ...
Necdet Batır, Hakan Kucuk, Sezer Sorgun
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