Results 31 to 40 of about 165,625 (255)
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
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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
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Stirling’s Approximation for Central Extended Binomial Coefficients [PDF]
The American Mathematical Monthly, 2014 Slight modification of journal version; title ...
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Series Containing Squared Central Binomial Coefficients and Alternating Harmonic Numbers [PDF]
Mediterranean Journal of Mathematics, 2019 The author presents an interesting integration technique in order to evaluate infinite sums containing harmonic or alternating harmonic numbers. He gives known as well as new results by applying his method. We quote the following formula: \[\sum^\infty_{n=1} \begin{pmatrix} 2n\\ n\end{pmatrix}^2\,H_{2n}/16^n(n+1)^2= \frac{16G+24-48\ln(2)}{\pi}+ 4-8\ln ...
J. Campbell
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Three pairs of congruences concerning sums of central binomial coefficients
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 ...
Guo-Shuai Mao, R. Tauraso
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Identities for squared central binomial coefficients
, 2021 We prove four identities for the squared central binomial coefficients. The first three of them reflect certain transformation properties of the complete elliptic integrals of the first and the second kind, while the last one is based on properties of the Lagrange polynomials.
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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 ...
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On an Asymptotic Series of Ramanujan [PDF]
, 2009 An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a function satisfying
A. Gut +19 more
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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|\
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Asymptotic expansion for inverse moments of binomial and Poisson distributions
, 2005 An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution.
Znidaric, Marko
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