Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0...Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.展开更多
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 go...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.展开更多
A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, comput...A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, computational search and symbolic dynamics, the mathematical structure underlying the time series generated from the elementary cellular automaton of rule 56 is analyzed and its complexity is determined, in which the Dyck language and Catalan numbers emerge naturally.展开更多
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...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 numbers of Dyck paths counted under different conditions, and f（n）, 9（n） and m are functions depending only on about n.展开更多
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...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.展开更多
A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and...A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.展开更多
Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0...Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0 respectively.In this paper,we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence.Then we prove the log-convexity of{Vn^2-V(n-1)V(n+1)}n≥2 and{n!Vn}n≥1,the ratio log-concavity of{Pn}n≥0 and the sequence{An}n≥0 of Apéry numbers,and the ratio log-convexity of{Vn}n≥1.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10871087)the Overseas Cooperation Fund of China (Grant No.10928101)
文摘Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.
文摘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.
基金This work is supported by the Special Funds for Major State Basic Research Project.
文摘A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, computational search and symbolic dynamics, the mathematical structure underlying the time series generated from the elementary cellular automaton of rule 56 is analyzed and its complexity is determined, in which the Dyck language and Catalan numbers emerge naturally.
基金the "973" Project on Mathematical Mechanizationthe National Science Foundation, the Ministry of Education, and the Ministry of Science and Technology of China.
文摘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 numbers of Dyck paths counted under different conditions, and f（n）, 9（n） and m are functions depending only on about n.
基金Project (No. 10371107) supported by the National Natural Science Foundation of China
文摘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.
文摘A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.
基金partially supported by the National Science Foundation of Xinjiang Uygur Autonomous Region(No. 2017D01C084)the National Science Foundation of China (Nos. 11771330 and 11701491)
文摘Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0 respectively.In this paper,we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence.Then we prove the log-convexity of{Vn^2-V(n-1)V(n+1)}n≥2 and{n!Vn}n≥1,the ratio log-concavity of{Pn}n≥0 and the sequence{An}n≥0 of Apéry numbers,and the ratio log-convexity of{Vn}n≥1.
