Irreducible polynomials over finite fields with prescribed trace/prescribed constant term

We obtain an equivalent version of Carlitzʹs formula for the number of monic irreducible polynomials of degree n and trace γ≠0 over a finite field via an integer recurrence. Similar expressions for the γ=0 case are also given. We also obtain formulas for the number of monic irreducible polynomials of degree n and prescribed constant term.