Research On The Matrix For Some Classical Combinatorial Sequences

We intend to investigate the relations between the classical combinatorial sequences and their matrices in this paper. They are listed as follows:1. The binomial coefficients and sequence an,k such that, aan,k=an-1,k-1+an-1,k are studied. The corresponding matrices and their properties are investigated. Some valuable combinatorial identities are derived.2. Lah number and its matrix Ln is studied. The relations among Lah matrices, Pascal matrices and Stirling matrices are obtained. The factorizations and power sum form on Lah matrices are given.3. The matrices with elements and their properties have been studied carefully by many mathematicians, where ψn(x) are xn,xn|λ and other special forms respectively. In this paper, the matrices Ln[x] = (l(i,j)ψi-j(x))n×n with more general elements are investigated, where l(n,k) and ψn(x) satisfy the following respectively:In fact, ψn(x) is the polynomial of binomial type. Therefore, the results we obtained are more general.4. Fibonacci sequences are of long history and their applications appear in many fields. The following bivariate Fibonacci sequences (polynomials) are studied:Some special cases of two-order linear recurrences are investigated and related combinatorial identities are given.A generalized Tribonacci matrix is constructed. The Tribonaccisequences {Tn} defined by: Tn+1= xTn+ yTn-1+zTn-2,T0=0,T1=1,T2=x are studied by the Tribonacci matrix. A relation between third-order linear recurrence and Fibonacci numbers is built, and some interesting expressions on Fibonacci numbers are derived.