Results 31 to 40 of about 97,613 (280)
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 ...
Mark R. Sepanski
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The p-adic valuation of k-central binomial coefficients [PDF]
Acta Arithmetica, 2009 11 ...
Armin Straub +2 more
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The Series of Reciprocals of Non-central Binomial Coefficients
American Journal of Computational Mathematics, 2013 Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients.
Laiping Zhang, JI Wan-hui
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On sums related to central binomial and trinomial coefficients [PDF]
, 2011 A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients.
Zhi‐Wei Sun
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On a conjecture of Graham on the p$p$‐divisibility of central binomial coefficients [PDF]
MathematikaAbstractLet be pairwise distinct primes. From a theorem of Kummer, each prime can divide at most times. We show that, for all , if are sufficiently large in terms of and , then there exist infinitely many positive integers such that each divides at most times.
Ernie Croot +2 more
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An identity for the central binomial coefficient
, 2012 We find the joint distribution of three simple statistics on lattice paths of n upsteps and n downsteps leading to a triple sum identity for the central binomial coefficient {2n}-choose-{n}. We explain why one of the constituent double sums counts the irreducible pairs of compositions considered by Bender et al., and we evaluate some of the other sums.
David Callan
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On congruences related to central binomial coefficients, harmonic and Lucasnumbers [PDF]
TURKISH JOURNAL OF MATHEMATICS, 2016 In this paper, using some combinatorial identities, we present new congruences involving central binomial coefficients and harmonic, Catalan, and Fibonacci numbers. For example, for an odd prime $p$, we have \begin{eqnarray*} \sum\limits_{k=1}^{\left( p-1\right) /2}\left( -1\right) ^{k}\binom{2k}{k}% H_{k-1} &\equiv &\frac{2^{p}}{p}\left( 2F_{p+1}-5 ...
Sibel Koparal, Neşe Ömür
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Three combinatorial sums involving central binomial coefficients [PDF]
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums.
Kunle Adegoke, Robert Frontczak
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A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence
, 2023 We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals generated from elements defined by polynomial recurrence relations, and we prove the conditions under which the set ...
Joseph Shunia
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Two permutation classes enumerated by the central binomial coefficients
, 2013 We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a bijection that allows us to determine some notable features of these permutations, such as the distribution of the ...
Marilena Barnabei +2 more
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