Results 61 to 70 of about 100,499 (167)
On Some Series Involving the Central Binomial Coefficients
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Adegoke, K., Frontczak, R., Goy, T.
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On Sums Related to Central Binomial and Trinomial Coefficients [PDF]
, 2014 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.
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Integral Representations of Catalan Numbers and Sums Involving Central Binomial Coefficients
, 2023 In the paper, the authors collect several integral representations of the Catalan numbers and central binomial coefficients, supply alternative proofs of two integral representations of the Catalan numbers, and apply these integral representations to alternatively prove several combinatorial identities for finite and infinite sums in which central ...
Guo, Bai-Ni, Lim, Dongkyu
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On statistical treatment of the results of parallel trails with special reference to fishery research [PDF]
, 1945 Parallel trials form a most important part of the technique of scientific experimentation. Such trials may be divided into two; categories. In the first the results are comparable measurements of one kind or another.
Buchanan-Wollaston, H.J.
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Three combinatorial sums involving central binomial coefficients
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.
Adegoke, Kunle, Frontczak, Robert
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Some $q$-congruences involving central $q$-binomial coefficients
, 2021 Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for all $d=0,1, \ldots n$ and $n= p^a$, where $a$ is a positive integer. In this paper, we present some $q$-analogues of
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Neuropsychiatric Disease and Treatment, 2016 Honglei Yin,1,2 Lin Xu,3 Yechang Shao,2,4 Liping Li,5 Chengsong Wan2 1Department of Psychiatry, Nanfang Hospital, Southern Medical University, Guangzhou, People’s Republic of China; 2School of Public Health, Southern Medical University ...
Yin HL, Xu L, Shao YC, Li LP, Wan CS
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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 ...
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On congruences related to central binomial coefficients, harmonic and Lucasnumbers
TURKISH JOURNAL OF MATHEMATICS, 2016 Summary: 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{aligned} \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_ ...
KOPARAL, SİBEL, ÖMÜR, NEŞE
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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.
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