LetG=Zpbe a cyclic group of prime orderpwith a representationG→GL(V) over a fieldKof characteristicp. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers ofVin the case thatVis indecomposable. From these they derived formulas for the Hilbert series of the invariant ringK[V]G. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers ofVinto indecomposables. Our results generalize to groups of the typeZp×Hwith |H| coprime top. Moreover, we prove that for any finite groupGwhose order is divisible bypbut not byp2the invariant ringA,K[V]Gis generated by homogeneous invariants of degrees at most dim (V).(|G| – 1).