  We compute the QCD corrections to the production of a top quark pair in
association with one hard jet at the Tevatron and the LHC, using the method of
generalized D-dimensional unitarity. Top quark decays are included at leading
order in perturbative QCD. We present kinematic distributions of top quark
decay products in lepton plus jets and dilepton final states at the Tevatron
and the LHC, using realistic selection cuts. We confirm a strong reduction of
the top quark forward-backward asymmetry for the process ttbar+jet at the
Tevatron at next-to-leading order, first observed by Dittmaier, Uwer and
Weinzierl. We argue that there is a natural way to understand this reduction
and that it does not imply a breakdown of the perturbative expansion for the
asymmetry.
