Results 11 to 20 of about 21,447 (236)
On a divisor of the central binomial coefficient [PDF]
The Ramanujan Journal, 2021 It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of the central binomial coefficient ${2n\choose n}$. Since the $n$th central binomial coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $n+ 1$ paths or
Matthew Just, Maxwell Schneider
openaire +2 more sources
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
openaire +2 more sources
Practical central binomial coefficients [PDF]
Quaestiones Mathematicae, 2020 A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient $\binom{2n}{n}$ is a practical number for all positive integers $n \leq x$ but at most $O(x^{0.88097})$ exceptions.
openaire +3 more sources
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 ...
openaire +2 more sources
Delannoy numbers and Legendre polytopes [PDF]
Discrete Mathematics & Theoretical Computer Science, 2008 We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial.
Gábor Hetyei
doaj +1 more source
Human and constructive proof of combinatorial identities: an example from Romik [PDF]
Discrete Mathematics & Theoretical Computer Science, 2005 It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the ...
D. Merlini, R. Sprugnoli, M. C. Verri
doaj +1 more source
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 $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}
TAURASO, ROBERTO, Sun, ZW
openaire +5 more sources
Journal of Water and Health, 2023 This study evaluated the results recorded at the Central Public Health Laboratory of Santa Catarina state (Brazil) concerning the investigation of Rotavirus (RVA) and Norovirus (NoVs) – genogroups GI and GII. Samples were taken from seawater, river water,
Andreza Mortari +6 more
doaj +1 more source
Stirling's Approximation for Central Extended Binomial Coefficients [PDF]
The American Mathematical Monthly, 2014 Slight modification of journal version; title ...
openaire +3 more sources
Products and Sums Divisible by Central Binomial Coefficients [PDF]
The Electronic Journal of Combinatorics, 2013 In this paper we study products and sums divisible by central binomial coefficients. We show that $$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|\
openaire +4 more sources