Results 11 to 20 of about 2,375 (206)
The 2-adic valuations of Stirling numbers of the second kind [PDF]
International Journal of Number Theory, 2009 In this paper, we investigate the 2-adic valuations of the Stirling numbers S(n, k) of the second kind. We show that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i ≢ 7 (mod 32). This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that v2(S(2n+ 1, k + 1)) = s2(n) - 1 for any positive integer n, where s2(n) is the sum ...
Shaofang Hong, Jianrong Zhao, Wei Zhao
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Applied Mathematics in Science and EngineeringIn this paper, we introduce the probabilistic Bernoulli numbers, Cauchy numbers, and Euler numbers of order α associated with the random variable Y, utilizing the generating function approach.
Aimin Xu
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On 2-Adic Orders of Stirling Numbers of the Second Kind [PDF]
, 2005 We prove that for any k = 1,... , 2n the 2-adic order of the Stirling number S(2n, k) of the second kind is exactly d(k) − 1, where d(k) denotes the number of 1’s among the binary digits of k. This confirms a conjecture of Lengyel.
Stefan De Wannemacker
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Multisectioned moments of stirling numbers of the second kind
Journal of Combinatorial Theory, Series A, 1973 The Stirling number of the second kind, S(n, k), enumerates the ways that n distinct objects can be stored in k non-empty indistinguishable boxes. When k is restricted to a given residue class modulo μ, the moments of the distribution S(n, k) have properties associated with the Olivier functions of order μ evaluated at 1 and −1. The simplest example is
D. H. Lehmer
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Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind
Journal of Approximation Theory, 2001 Let \(Q_n(z) = \sum_{k=1}^n S(n,k) (nz)^k,\) where \(S(n,k)\) is the Stirling number of the second kind. The author proves an asymptotic formula for \(Q_n(z)\) with \(z \in {\mathbf C}\backslash[-e,0]\) and proves an Airy-asymptotic formula for \(Q_n\) as \(z \to -e\).
Christian Elbert
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p-adic Stirling numbers of the second kind
The Fibonacci Quarterly, 2013 13 ...
Donald M. Davis
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Some results on p-adic valuations of Stirling numbers of the second kind
AIMS Mathematics, 2020 Let $n$ and $k$ be nonnegative integers. The Stirling number of the second kind, denoted by $S(n, k)$, is defined as the number of ways to partition a set of $n$ elements into exactly $k$ nonempty subsets and we have $$ S(n, k)=\frac{1}{k!}\sum_{i=0}^{k}(
Yulu Feng, Min Qiu
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On the asymptotic normality of the Legendre–Stirling numbers of the second kind
European Journal of Combinatorics, 2015 For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling ...
Wolfgang Gawronski +2 more
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Multi-Stirling numbers of the second kind
FilomatThe multi-Stirling numbers of the second kind, the unsigned multi-Stirling numbers of the first kind, the multi-Lah numbers and the multi-Bernoulli numbers are all defined with the help of the multiple logarithm, and generalize respectively the Stirling numbers of the second kind, the unsigned Stirling numbers of the first kind, the unsigned Lah ...
Taekyun Kim, Dae San Kim, Hye Jin Kim
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Stirling numbers of the second kind and Bonferroni's inequalities
Elemente der Mathematik, 2005 Horst Wegner
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