We consider the ring of invariants RG obtained by letting a finite group G act via its regular representation on a polynomial ring withindeterminates over an algebraically closed field of characteristic zero. We note that the Hilbert series of RG can be thought of as a function of a certain sequence, namely the order structure of G. We consider properties of this and a related function for more general classes of sequences. As a result of our findings, we use invariant theory to give new proofs of two theorems, both originally due to Probenius, on finite groups.