Results 211 to 220 of about 24,612 (239)
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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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Summation Formulas on Harmonic Numbers and Five Central Binomial Coefficients
Mathematical Notes, 2023zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Li, Chunli, Chu, Wenchang
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Solution to a problem involving central binomial coefficients
Integral Transforms and Special Functionsexaly +2 more sources
INFINITE SERIES WITH HARMONIC NUMBERS AND CENTRAL BINOMIAL COEFFICIENTS
International Journal of Number Theory, 2009By 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, Deyin
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PROOF OF TWO CONJECTURES ON SUPERCONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS
Bulletin of the Australian Mathematical Society, 2020In this note we use some $q$-congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
CHENG-YANG GU, VICTOR J. W. GUO
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ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS
Bulletin of the Australian Mathematical SocietyAbstractWe prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory13(11) (2011), 2219–2238]. Let p be an odd prime. Then $$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$ where $H_n$ is the nth harmonic
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A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients
The American Mathematical Monthly, 2014We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain.
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On certain sums divisible by the central binomial coefficient
J. Integer Seq., 2020Summary: We prove that some sums, which arise as generalizations of known binomial coefficient identities, are divisible by the central binomial coefficient. A new method is used. In particular, we show that an alternating sum concerning the product of a power of a binomial coefficient with two Catalan numbers is always divisible by the central ...
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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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