The number of chains of subgroups of a finite cyclic group

T?rn?uceanu and Bentea M. T?rn?uceanu, 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 T?rn?uceanu M. T?rn?uceanu, 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 T?rn?uceanu and Bentea.