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Congruences for finite triple harmonic sums

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2007年第6期8卷 946-948页

作者： FU Xu-dan ZHOU Xia CAI Tian-xin Department of Mathematics Zhejiang University Hangzhou 310028 China Hangzhou Foreign Language School Hangzhou 310023 China

Zhao （2003a） first established a congruence for any odd prime p〉3, S（1,1,1 ;p）=-2Bp-3 （mod p）, which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S（α,β, γ,ρ）（modp） is considered for all positive integers α,β, γ. We refer to w=α＋β＋ γ as the weight of the sum, and show that if w is even, S（α,β, γ,ρ）=0 （mod p） for p≥w＋3; if w is odd, S（α,β, γ,ρ）=-rBp-w （mod p） for p≥w, here r is an explicit rational number independent ofp. A congruence of catalan number is obtained as a special case.

关键词： Finite triple harmonic sums Recursive relation Bernoulli numbers catalan numbers

Powers of the catalan Generating Function and Lagrange's 1770 Trinomial Equation Series

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Journal of Mathematical Research with Applications 2019年第6期39卷 603-606页

作者： H.W.GOULD Department of Mathematics West Virginia University

The catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,... are given by C（n）=■for n ≥ 0. They are named for Eugene catalan who studied them as early as *** were also found by Leonhard Euler（1758）, Nicholas von Fuss（1795）, and Andreas von Segner（1758）. The catalan numbers have the binomial generating function ■It is known that powers of the generating function C（z） are given by ■The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation x-3x + 1 = 0, the numbers ■analogous to catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.

关键词： catalan numbers Vandermonde convolution Lagrange and Bürmann series Rothe’s formula(or general identity of Rothe-Hagen)

Counting Dyck Paths with Strictly Increasing Peak Sequences

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Journal of Mathematical Research and Exposition 2007年第2期27卷 253-263页

作者：孙怡东贾藏芝大连理工大学应用数学系辽宁大连116024

In this paper we consider the enumeration of subsets of the set, say Dm, of those Dyck paths of arbitrary length with maximum peak height equal to m and having a strictly increasing sequence of peak height （as one goes along the path）. Bijections and the methods of generating trees together with those of Riordan arrays are used to enumerate these subsets, resulting in many combinatorial structures counted by such well-known sequences as the catalan nos., Narayana nos., Motzkin nos., Fibonacci nos., Schroeder nos., and the unsigned Stirling numbers of the first kind. In particular, we give two configurations which do not appear in Stanley＇s well-known list of catalan structures.

关键词： Generating tree Riordan array catalan numbers Schroeder numbers.

Identities Involoving Some numbers Related to Dyck Paths

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Journal of Mathematical Research and Exposition 2005年第3期25卷 441-446页

作者：孙怡东南开大学组合数学研究中心天津300071

In the paper we derive many identities of forms ∑i=0^n（-1）^n-i（i^n）Um＋k＋i,k＋i=f（n）and ∑ i=o^2n（-1）^i（i^2n）Um＋k＋i,k＋i=9（n）by the Cauchy Residue Theorem and an operator method, where Un, k are number... 详细信息

关键词： Identities Dyck path catalan numbers Motzkin numbers.