Results 21 to 30 of about 18,619 (166)

Series associated with harmonic numbers, Fibonacci numbers and central binomial coefficients $binom{2n}{n}$ [PDF]

Notes on Number Theory and Discrete Mathematics
We find various series that involve the central binomial coefficients $binom{2n}{n}$, harmonic numbers and Fibonacci numbers. Contrary to the traditional hypergeometric function _pF_q approach, our method utilizes a straightforward transformation to ...
Segun Olofin Akerele, Olamide Esther Salami   +1 more
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Alternating Convolutions of Catalan Numbers [PDF]

Bulletin of the Brazilian Mathematical Society, New Series, 2021
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
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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 ...
Feng Qi, Bai-Ni Guo
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Growing and Destroying Catalan-Stanley Trees [PDF]

Discrete Mathematics & Theoretical Computer Science, 2018
Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers.
Benjamin Hackl, Helmut Prodinger
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Simplifying coefficients in differential equations for generating function of Catalan numbers

Journal of Taibah University for Science, 2019
In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients in two families of nonlinear ordinary differential equations for the generating ...
Feng Qi, Yong-Hong Yao
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On a new congruence in the Catalan triangle [PDF]

Notes on Number Theory and Discrete Mathematics
For 0≤k≤n, the number C(n,k) represents the number of all lattice paths in the plane from the point (0,0) to the point (n,k), using steps (1,0) and (0,1), that never rise above the main diagonal y = x.
Jovan Mikić
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On the $H$-triangle of generalised nonnesting partitions [PDF]

Discrete Mathematics & Theoretical Computer Science, 2014
With a crystallographic root system $\Phi$ , there are associated two Catalan objects, the set of nonnesting partitions $NN(\Phi)$, and the cluster complex $\Delta (\Phi)$. These possess a number of enumerative coincidences, many of which are captured in
Marko Thiel
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Chromatic statistics for Catalan and Fuß-Catalan numbers

Electronic Journal of Combinatorics, 2011
We refine Catalan numbers and Fuß-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fuß-Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively Fuß-Catalan complexes, with a given colour distribution of its vertices is ...
Bacher, Roland, Krattenthaler, Christian
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Volterra Chain and Catalan Numbers [PDF]

JETP Letters, 2018
4 pages, 2 ...
Adler, V. E., Shabat, A. B.
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Generalized Catalan Numbers from Hypergraphs [PDF]

The Electronic Journal of Combinatorics, 2021
The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions.
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