The number of chains of subgroups of a finite cyclic group

Tărnăuceanu and Bentea [M. Tărnăuceanu, 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 Tărnăuceanu [M. Tărnăuceanu, 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 Tărnăuceanu and Bentea.