There is a longstanding conjecture by Fröberg about the Hilbert series of the ring R∕I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m’th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.