scholarly journals Divisibility of the central binomial coefficient $\binom {2n}{n}$

2020 ◽  
Vol 374 (2) ◽  
pp. 923-953
Author(s):  
Kevin Ford ◽  
Sergei Konyagin
Keyword(s):  
Binomial Coefficient ◽  
Central Binomial Coefficient

scholarly journals INTEGRAL REPRESENTATIONS AND INEQUALITIES OF EXTENDED CENTRAL BINOMIAL COEFFICIENTS

2021 ◽  
Author(s):  
Chunfu Wei
Keyword(s):  
Binomial Coefficient ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Exponential Functions ◽  
Central Binomial Coefficient

In the paper, the author presents three integral representations of extended central binomial coefficient, proves decreasing and increasing properties of two power-exponential functions involving extended (central) binomial coefficients, derives several double inequalities for bounding extended (central) binomial coefficient, and compares with known results.

scholarly journals Asymptotic approximation of central binomial coefficients with rigorous error bounds

2021 ◽  
Vol 5 (1) ◽  
pp. 380-386
Author(s):  
Richard P. Brent ◽  
Keyword(s):  
Theta Function ◽  
Error Bounds ◽  
Asymptotic Approximation ◽  
Asymptotic Series ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Rigorous Error Bounds ◽  
Historical Remarks ◽  
Rigorous Error ◽  
Central Binomial Coefficient

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.

scholarly journals Central binomial coefficients divisible by or coprime to their indices

2018 ◽  
Vol 14 (04) ◽  
pp. 1135-1141 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Upper Density ◽  
Positive Integers ◽  
Central Binomial Coefficient

Let [Formula: see text] be the set of all positive integers [Formula: see text] such that [Formula: see text] divides the central binomial coefficient [Formula: see text]. Pomerance proved that the upper density of [Formula: see text] is at most [Formula: see text]. We improve this bound to [Formula: see text]. Moreover, let [Formula: see text] be the set of all positive integers [Formula: see text] such that [Formula: see text] and [Formula: see text] are relatively prime. We show that [Formula: see text] for all [Formula: see text].

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