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A Simple Probabilistic Proof for the Alternating Convolution of the Central Binomial Coefficients

American Statistician, 2018
This note presents a simple probabilistic proof of the identity for the alternating convolution of the central binomial coefficients. The proof of the identity involves the computation of moments of order n for the product of standard normal random ...
A. K. Pathak
semanticscholar   +1 more source

INFINITE SERIES WITH HARMONIC NUMBERS AND CENTRAL BINOMIAL COEFFICIENTS

International Journal of Number Theory, 2009
By means of two hypergeometric summation formulae, we establish two large classes of infinite series identities with harmonic numbers and central binomial coefficients. Up to now, these numerous formulae have hidden behind very few known identities of Apéry-like series for Riemann-zeta function, discovered mainly by Lehmer [14] and Elsner [12] as well ...
CHU, Wenchang, ZHENG D. Y.
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Powers in prime bases and a problem on central binomial coefficients

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0.
Sebastian Tim Holdum, Frederik Ravn Klausen, P. Rasmussen   +2 more
semanticscholar   +1 more source

Interesting Series Involving the Central Binomial Coefficient

The American Mathematical Monthly, 1985
(1985). Interesting Series Involving the Central Binomial Coefficient. The American Mathematical Monthly: Vol. 92, No. 7, pp. 449-457.
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Wallis's Product and the Central Binomial Coefficient

The American Mathematical Monthly, 2015
(2015). Wallis's Product and the Central Binomial Coefficient. The American Mathematical Monthly: Vol. 122, No. 7, pp. 689-689.
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DIVISIBILITY OF CERTAIN SUMS INVOLVING CENTRAL q-BINOMIAL COEFFICIENTS

Rocky Mountain Journal of Mathematics
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Chen, Yifan, Wang, Xiaoxia
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Integral representations and properties of several finite sums containing central binomial coefficients

ScienceAsia, 2023
Feng Qi (祁锋), D. Lim
semanticscholar   +1 more source

Some congruences involving fourth powers of central q-binomial coefficients

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2019
AbstractWe prove some congruences on sums involving fourth powers of central q-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]: $$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {
Guo, Victor J. W., Wang, Su-Dan
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Number of divisors of the central binomial coefficient

Moscow University Mathematics Bulletin, 2013
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Brain and other central nervous system tumor statistics, 2021

Ca-A Cancer Journal for Clinicians, 2021
Kimberly D Miller, Quinn T Ostrom, Carol Kruchko Ba   +2 more
exaly