Counting non-isomorphic finite nilpotent groups of order n is a very hard problem. One way to approach this problem is to count finite nilpotent groups of fixed nilpotency class c on d generators. The enumeration of such isomorphism classes of objects involves number theory and the theory of algebraic groups. However, very little is known about the explicit generating functions of these sequences of numbers when c > 2 or d > 2. We use a direct enumeration of such groups that began in the works of M. Bacon, L. Kappe, et al, to provide a natural multivariable extension of the generating function counting such groups. Then we rederive the explicit formulas that are known so far.