Results 21 to 30 of about 44,005 (262)
On the divisibility of binomial coefficients
Ars Mathematica Contemporanea, 2020 In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is divisible by at least one of $p$ or $q$. We give conditions under which a number $n$ has this property and discuss a
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On Binomial Coefficient Residues [PDF]
Canadian Journal of Mathematics, 1957 The number of binomial coefficients , which are congruent to j , 0 ≤ j ≤ p − 1, modulo the prime number p is denoted by θj(n). In this paper we give systems of simultaneous linear difference equations with constant coefficients whose
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Sums of Reciprocals of Triple Binomial Coefficients
International Journal of Mathematics and Mathematical Sciences, 2008 We investigate the integral representation of infinite sums involving the reciprocals of triple binomial coefficients. We also recover some wellknown properties of 𝜁(3) and extend the range of results given by other authors.
A. Sofo
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On some series involving the binomial coefficients $binom{3n}{n}$ [PDF]
Notes on Number Theory and Discrete MathematicsUsing a simple transformation, we obtain much simpler forms for some series involving binomial coefficients $binom{3n}{n}$ derived by Necdet Batir. New evaluations are given and connections with Fibonacci numbers and the golden ratio are established ...
Kunle Adegoke +2 more
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Poset binomials and rainbow characters [PDF]
Discrete Mathematics & Theoretical Computer Science, 2013 This paper introduces a variation on the binomial coefficient that depends on a poset and interpolates between $q$-binomials and 1-binomials: a total order gives the usual $q$-binomial, and a poset with no relations gives the usual binomial coefficient ...
Daniel Bragg, Nathaniel Thiem
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A sum of binomial coefficients [PDF]
Mathematics of Computation, 1978 An explicit expression is derived for the sum of the ( k + 1 ) (k + 1) st binomial coefficients in the nth, ( n − m ) (n - m) th, ( n − 2 m ) (n - 2m) th,... row of the arithmetic triangle.
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Ray trajectories, binomial coefficients of a new type, and the binary system [PDF]
Компьютерные исследования и моделирование, 2010 The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be binomial coefficients
Aleksandr Vladimirovich Yurkin
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Plane Partitions and a Problem of Josephus
Mathematics, 2023 The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the position ...
Mircea Merca
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Inequalities for Binomial Coefficients
Journal of Mathematical Analysis and Applications, 1999 For any real number \(r\) with \(r>1\), let \(c_r= (2\pi(1-{1\over r}))^{-1/2}\) and \(d_r= (r-1)/(1-{1\over r})^r\). Let \(B_{2m}\) \((m= 1,2,\dots)\) be the Bernoulli numbers defined by \[ {z\over e^z-1}=1-{z\over 2}+\sum^\infty_{m=1} B_{2m}{z^{2m}\over (2m)!}.
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