On The Central Coefficients Of Riordan Matrices And Its Applications

A Riordan array is an infinite, lower triangular array. In our thesis, we charac-terize the central coefficients of Riordan matrices in four subgroups of the Riordan group, that is, the Bell subgroup, the associated subgroup, the derivative subgroup, the hitting time subgroup. In addition, we stretch the triangle so that it becomes isosceles. Considering the column of the right part of the ISO triangle, we define the r-shifted central coefficients, which is beneficial to the combinatorics.Using the Lagrange Inversion Formula, we give the r-shifted central coefficients computing methods in four subgroups. Some examples are also presented to show that how we deduce the generating functions for interesting sequences by using different means of calculating the r-shifted central coefficients. Moreover, we find some combinatorial applications by the two-stage weighted lattice paths. At last, some extensions are shown.