Results 31 to 40 of about 160,421 (323)
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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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.
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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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A generalization of the binomial coefficients
Discrete Mathematics, 1992 We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici d terminer est la meilleure g n ralisation possible des factorielles et des coefficients du bin ome.
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On congruences for binomial coefficients
Journal of Number Theory, 1989 For primes \(p=1+4f=a^ 2+b^ 2\), \(a\equiv 1 (mod 4)\), Gauss proved that \(\left( \begin{matrix} 2f\\ f\end{matrix} \right)\equiv 2a (mod p)\). For primes \(p=1+kf\) (k\(\geq 3)\), many congruences modulo p, for the binomial coefficients \(\left( \begin{matrix} rf\\ sf\end{matrix} \right ...
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The update exposition of the components organising the sum of weighted equal powers
Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki, 2012 The sum of the weighted equal powers with natural bases and parameters is organized of components, which are independent or dependent on weight coefficients.
A. I. Nikonov
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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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Some congruences involving binomial coefficients
, 2015 Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant ...
Cao, Hui-Qin, Sun, Zhi-Wei
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Reduction of the sum of the weight equal powers to explicit combinatorial representation
Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki, 2012 The paper contains the proof of the statement that the component of the sum of weighted powers with natural bases and equal parameters, dependent on weight coefficients, is equal to the sum of products of binomial and weight coefficients.
A. I. Nikonov
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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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