Moment formulas for the quasi-nilpotent DT-operator

Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated R-transform we compute the moments of (T - lambda1)* (T - lambda1) where lambda is an element of C, and the Brownmeasure of T + rootis an element ofY, where Y is a circular element *-free from T for is an element of > 0. Moreover we give a new proof of Sniady's formula for the moments tau(((T*)T-k(k))(n)) for k, n is an element of N.