A Class Of Two-term Nonlinear Recurrence Relation And Its Related Combinatorial Sequences

Recurrence relation is one of the most classical and major research directions in combinatorics.Therefore,it has attracted the attentions of many combinatorial scholars at home and abroad.Generally,recurrence relation can be divided into linear recurrence relation and nonlinear recurrence relation.In this paper,we focus on a class of two-term nonlinear recurrence relation and its related combinatorial sequences.First of all,this paper discusses the general solution of homogeneous and nonhomogeneous linear recurrence relation with constant coefficients.Furthermore,three combinatorial sequences which satisfy recurrence relation with constant coefficients,including derangement numbers,Fibonacci numbers and Pell numbers,and their properties are studied.Secondly,in this paper,we generalize a class of two-term nonlinear recurrence relation,which proposed by Graham,Knuth and Patashnik in ‘concrete mathematics'.By using the weighted combinatorial model,we give the general solution of this class of recurrence relation.Moreover,by virtue of symmetric functions,the explicit formulas for four special cases of this class of recurrence relation are also provided.Finally,in this paper,six combinatorial sequences,including two types of Eulerian numbers,Lah numbers,q-Lah numbers,r-Whitney-Lah numbers and(p,q)-Whitney Lah numbers,and their properties are studied.These combinatorial sequences are all special cases of the generalized two-term nonlinear recurrence relation.