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
Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means ...Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas. 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 two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any...In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any p, sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p = 4, 5. The present paper presents the desired expansion for arbitrary integer p.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
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
The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <i...The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <img src="Edit_909dc64a-717a-4477-a9f8-a3b94ab4008e.bmp" alt="" /> for representing the zeta function at the odd integer values 2<em>n</em>+1 for <em>n</em> a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers <em>E</em><sub>2<em>n</em></sub> and the polygamma functions <em>ψ</em><sup>(2<em>n</em>)</sup>(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.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
The judicious choice of metal clusters and organic building blocks leads to a wide variety of structures for metal–organic frameworks(MOFs).In this work,we demonstrated that relegating the symmetry of a building bloc...The judicious choice of metal clusters and organic building blocks leads to a wide variety of structures for metal–organic frameworks(MOFs).In this work,we demonstrated that relegating the symmetry of a building block can also lead to the proliferation of new MOF structures.Herein,a triangle building block was elongated with reduced symmetry for MOF construction,which gave rise to a novel(3,4,6)-connected idp(based on the definition of Reticular Chemistry Structure Resource,http://rcsr.net/)network(PFC-16)with mesopores and abundant open-metal sites.The framework is composed of the rarely observed tetrakis hexahedral cages,surrounding which are small cages arranged in sodalite topology.The relegated symmetry was required for this novel self-assembly.The obtained MOF with mesopores,a robust backbone,and abundant open-metal sites can incorporate functional species in the structure,which is representatively demonstrated by internalizing the photosensitizer zinc(II)phthalocyanine(ZnPc)and the modifying tumor-targeting molecule folic acid(FA)in PFC-16.The obtained composite FA-ZnPc@nano-PFC-16 shows excellent photodynamic therapy(PDT)efficiency for both in vitro and in vivo experiments,representing a promising candidate for cancer therapy.展开更多
Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinaliti...Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinalities of the sets Dn(T) where T■B2.Some of the cardinalities encountered involve inverse binomial coefficients,binomial coefficients,Catalan numbers,and Fibonacci numbers.展开更多
The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-ser...The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).展开更多
In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of th...In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of the first kind,we found an alternative proof of a result of Ramanujan for3F2,a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.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Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.
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.
文摘In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any p, sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p = 4, 5. The present paper presents the desired expansion for arbitrary integer p.
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.
文摘The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <img src="Edit_909dc64a-717a-4477-a9f8-a3b94ab4008e.bmp" alt="" /> for representing the zeta function at the odd integer values 2<em>n</em>+1 for <em>n</em> a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers <em>E</em><sub>2<em>n</em></sub> and the polygamma functions <em>ψ</em><sup>(2<em>n</em>)</sup>(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.
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.
基金Authors are grateful for the support from National Key Research and Development Program of China(nos.2017YFA0206801 and 2018YFA0208600)National Natural Science Foundation of China(nos.21520102001 and 21871267)the Strategic Priority Research Program of the Chinese Academy of Sciences(no.XDB20000000).
文摘The judicious choice of metal clusters and organic building blocks leads to a wide variety of structures for metal–organic frameworks(MOFs).In this work,we demonstrated that relegating the symmetry of a building block can also lead to the proliferation of new MOF structures.Herein,a triangle building block was elongated with reduced symmetry for MOF construction,which gave rise to a novel(3,4,6)-connected idp(based on the definition of Reticular Chemistry Structure Resource,http://rcsr.net/)network(PFC-16)with mesopores and abundant open-metal sites.The framework is composed of the rarely observed tetrakis hexahedral cages,surrounding which are small cages arranged in sodalite topology.The relegated symmetry was required for this novel self-assembly.The obtained MOF with mesopores,a robust backbone,and abundant open-metal sites can incorporate functional species in the structure,which is representatively demonstrated by internalizing the photosensitizer zinc(II)phthalocyanine(ZnPc)and the modifying tumor-targeting molecule folic acid(FA)in PFC-16.The obtained composite FA-ZnPc@nano-PFC-16 shows excellent photodynamic therapy(PDT)efficiency for both in vitro and in vivo experiments,representing a promising candidate for cancer therapy.
基金The National Natural Science Foundation of China （No. 10801020）.
文摘Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinalities of the sets Dn(T) where T■B2.Some of the cardinalities encountered involve inverse binomial coefficients,binomial coefficients,Catalan numbers,and Fibonacci numbers.
文摘The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).
文摘In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of the first kind,we found an alternative proof of a result of Ramanujan for3F2,a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.
