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Some combinatorial identities containing central binomial coefficients or Catalan numbers* [PDF]

Applied Mathematics in Science and Engineering, 2023
In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; establish four ...
Feng Qi, Da‐Wei Niu, Dongkyu Lim
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Some congruences involving central q-binomial coefficients [PDF]

Advances in Applied Mathematics, 2010
16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv.
Victor J.W. Guo, Jiang Zeng
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Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function

Results in Nonlinear Analysis, 2021
In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are related to the ...
Feng Qi, Chao-ping Chen, Dongkyu Lim
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Factors of certain sums involving central q-binomial coefficients [PDF]

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo.
Victor J. W. Guo, Su-Dan Wang
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Practical central binomial coefficients [PDF]

Quaestiones Mathematicae, 2020
A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions.
Carlo Sanna
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Some $q$-congruences involving central $q$-binomial coefficients [PDF]

, 2021
Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for all $d=0,1, \ldots n$ and $n= p^a$, where $a$ is a positive integer. In this paper, we present some $q$-analogues of
He-Xia Ni
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Harmonic Series of Convergence Ratio “1/4" with Cubic Central Binomial Coefficients [PDF]

Axioms
We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “1/4”) containing the cubic central binomial coefficients and harmonic numbers is ...
Chunli Li, Wenchang Chu
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Sums of Reciprocals of the Central Binomial Coefficients

, 2006
We consider a set of combinatorial sums involving the reciprocals of the central binomial coefficients and try to solve (or close) them by means of generating functions. We obtain a number of results for infinite sums, in some of which the golden ratio ! appears.
Renzo Sprugnoli
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On Some Series Involving the Central Binomial Coefficients [PDF]

15 ...
Kunle Adegoke, Robert Frontczak, Taras Goy   +2 more
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Identities for squared central binomial coefficients [PDF]

, 2021
We prove four identities for the squared central binomial coefficients. The first three of them reflect certain transformation properties of the complete elliptic integrals of the first and the second kind, while the last one is based on properties of the Lagrange polynomials.
Khristo N. Boyadzhiev
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