A further generalization of the Catalan numbers and its explicit formula and integral representation

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

A further generalization of the Catalan numbers and its explicit formula and integral representation

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

Solutions to All-Colors Problem on Graph Cellular Automata

The All-Ones Problem comes from the theory of σ+-automata, which is related to graph dynamical systems as well as the Odd Set Problem in linear decoding. In this paper, we further study and compute the solutions to the “All-Colors Problem,” a generalization of “All-Ones Problem,” on some interesting classes of graphs which can be divided into two subproblems: Strong-All-Colors Problem and Weak-All-Colors Problem, respectively. We also introduce a new kind of All-Colors Problem, k-Random Weak-All-Colors Problem, which is relevant to both combinatorial number theory and cellular automata theory.

On Normal Sequence in Abelian Group Cn⊕Cn

Electronic information technology is based on expression, storage, transmission, and process of information. Information expression is usually generalized as information coding technology, one of whose heart theories is combinatorial number theory in algebra. In the process of storing, transmitting, and processing of electronic information technology, problems about encryption and safety need to use algebra theories of group, ring, and domain. It is thus clear that algebra is very important in electronic information theory. This paper makes use of abelian group basic theory of algebra, together with combination number theory, discusses construction problem of normal sequence in of abelian group, and gives several sufficient conditions for a guess establishment of W.D.Gao.