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 $c$.
Kevin Ford, Sergeĭ Konyagin
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Spatiotemporal Trends and Co-Resistance Patterns of Multidrug-Resistant Enteric Escherichia coli O157 Infections in Humans in the United States [PDF]
PathogensMultidrug-resistant (MDR) Shiga toxin-producing Escherichia coli O157 (STEC O157) is a public health threat. This study analyzed publicly available surveillance data collected by the National Antimicrobial Resistance Monitoring System (NARMS) to assess ...
Tarjani Bhatt, Csaba Varga
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Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers [PDF]
Symmetry, 2021 We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.
Necdet Batır, Anthony Sofo
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Odd Euler Sums and Harmonic Series with Cubic Central Binomial Coefficients in Denominators
AxiomsBy means of the coefficient extraction method, we examine a transformation of a classical hypergeometric series. Three classes of infinite series (of convergence rate “1/4”) with harmonic numbers in numerators and cubic central binomial coefficients in ...
Chunli Li, Wenchang Chu
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Congruences involving the reciprocals of central binomial coefficients
, 2009 We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.
Roberto Tauraso
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Overlapping genes connect rheumatoid arthritis and head and neck cancer: coincidence or shared immune pathophysiology? [PDF]
Frontiers in MedicineIntroductionDespite advances in understanding the pathophysiology of rheumatoid arthritis (RA) and head and neck cancer (HNC) individually, their shared genetic and molecular mechanisms remain poorly defined.MethodsThis study aimed to explore gene-level ...
Ran Wang +6 more
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Three pairs of congruences concerning sums of central binomial coefficients [PDF]
International Journal of Number Theory, 2021 Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula ...
Guo-Shuai Mao, Roberto Tauraso
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Some congruences involving central q-binomial coefficients
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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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)}
Zhi‐Wei Sun, Roberto Tauraso
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