Some Inverse Relations Determined by Catalan Matrices

We use the -sequence and -sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced. 1. Introduction The Catalan numbers have been widely encountered and investigated [1, 2]. They can be defined through binomial coefficients or by the generating function being which satisfies the functional equation . In [2], Stanley listed 66 enumerative problems which are counted by the Catalan numbers. Many number triangles related to the Catalan sequence have been introduced in the literature. In [3–5], Shapiro et al. introduced a Catalan triangle with the entries given by The following identity is obtained in [6] in connection with the moment of the Catalan triangle: Another proof of the above identity is given by Woan et al. [7] while computing the areas of parallelo-polyominoes via generating functions. In [8], a combinatorial interpretation of the matrix identity (4) is also obtained. In [9], Radoux introduced a triangle of numbers and he presents the identity with , which is equivalent to following matrix equation: Deng and Yan [10] proved this identity by using the Riordan array method. Aigner [11] introduced a number triangle with the entries given by This array is also discussed in [12–14]. We use the -sequence and -sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients, which are generalizations of (4) and (6). In addition, a unified form of Catalan matrices is introduced. 2. Riordan Arrays In the recent literature, one may find that Riordan arrays have attracted the attention of various authors from many points of view, and many examples and applications can be found (see, e.g., [13, 15–21]). An infinite lower triangular matrix is called a Riordan array if its column has generating function , where and are formal power series with , , and . The Riordan array is denoted by . Thus, the general term of Riordan array is given by where denotes the coefficient of in power series . Suppose we multiply the array by a column vector and get a column vector . Let be the ordinary generating function for the sequence . Then it follows that the ordinary generating function for the sequence is . If we identify a