Results 21 to 30 of about 7,070,887 (229)
Divisibility of the central binomial coefficient $\binom {2n}{n}$ [PDF]
Transactions of the American Mathematical Society, 2020 We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$.
Ford, Kevin, Konyagin, Sergei
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Sum of Successive Partitions of Binomial Coefficient
, 2022 This paper focuses on the successive partition method applied to a binomial coefficient in combinatorial geometric series. The coefficient for each term in combinatorial geometric series refers to a binomial coefficient.
Chinnaraji Annamalai
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On Divisibility of Convolutions of Central Binomial Coefficients [PDF]
The Electronic Journal of Combinatorics, 2014 Recently, Z. Sun proved that \[ 2(2m+1)\binom{2m}{m} \mid \binom{6m}{3m}\binom{3m}{m} \] for $m\in\mathbb{Z}_{>0}$. In this paper, we consider a generalization of this result by defining \[ b_{n,k}=\frac{2^{k}\, (n+2k-2)!!}{((n-2)!!\, k!}. \] In this notation, Sun's result may be expressed as $2\, (2m+1) \mid b_{(2m+1),(2m+1)-1}$ for $m\in\mathbb ...
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Clinical associations of Human T-Lymphotropic Virus type 1 infection in an indigenous Australian population. [PDF]
PLoS Neglected Tropical Diseases, 2014 INTRODUCTION: In resource-poor areas, infectious diseases may be important causes of morbidity among individuals infected with the Human T-Lymphotropic Virus type 1 (HTLV-1).
Lloyd Einsiedel +5 more
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New congruences for central binomial coefficients
Advances in Applied Mathematics, 2010 Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}
TAURASO, ROBERTO, Sun, ZW
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Prenatal factors associated with fetal visceral adiposity
Jornal de Pediatria (Versão em Português), 2020 Objective: To assess fetal visceral adiposity and associated factors during pregnancy. Methods: Secondary analysis of prospective cohort data with 172 pairs (pregnant woman/fetus) treated at public health units.
Aline Silva Santos Sena +5 more
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Stirling's Approximation for Central Extended Binomial Coefficients [PDF]
The American Mathematical Monthly, 2014 Slight modification of journal version; title ...
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Factors of certain sums involving central q-binomial coefficients [PDF]
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021 Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo.
Guo, Victor J. W., Wang, Su-Dan
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Products and Sums Divisible by Central Binomial Coefficients [PDF]
The Electronic Journal of Combinatorics, 2013 In this paper we study products and sums divisible by central binomial coefficients. We show that $$2(2n+1)\binom{2n}n\ \bigg|\ \binom{6n}{3n}\binom{3n}n\ \ \mbox{for all}\ n=1,2,3,\ldots.$$ Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k\ \bigg|\ \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k\ \bigg|\
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Ascending and Descending Orders of Annamalai’s Binomial Coefficient
, 2022 This paper presents the analysis of ascending and descending orders with Annamalai’s binomial coefficient compared with traditional combination of combinatorics.
Chinnaraji Annamalai
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