The number of chains of subgroups of a finite cyclic group

Tarnauceanu and Bentea [M. Tarnauceanu, L Bentea, On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems 159 (2008) 1084-1096] gave an explicit formula for the number of chains of subgroups in the lattice of a finite cyclic group by finding its generating function of one variable. Using this result Tarnauceanu [M. Tarnauceanu, Fuzzy subgroups of finite cyclic groups and Delannoy numbers, European J. Combin. 30 (2009) 283-287] found an explicit formula for the central Delannoy number. In this note we find a generating function of multi-variables for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group. As results we give simplified formulas for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group and for the central Delannoy numbers compared with the formulas given by Tarnauceanu and Bentea. (C) 2011 Elsevier Ltd. All rights reserved.