In this paper, we present some results on the distribution of dividend payments until ruin under a Sparre Andersen risk model with generalized Erlang(n)-distributed inter-claim times and a constant dividend barrier. An integro-differential equation for the moment-generating function of the sum of the discounted dividend payments until ruin is derived. Moreover, explicit solutions for arbitrary moments of the present value of dividend payments are obtained, when the individual claim amounts have a distribution with rational Laplace transform. Numerical illustrations of the results are given for an Erlang(2) risk model and Erlang(2)-distributed claim amounts.