  Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated
R-transform we compute the moments of $(T-\lambda 1)^*(T-\lambda 1)$, where
$\lambda \in \mathbb C$, and the Brown-measure of $T+\sqrt{\epsilon} Y$, where
Y is a circular element *-free from T for $\epsilon>0$. Moreover we give a new
proof of \'Sniady's formula for the moments $\tau(((T^*)^k T^k)^n)$ for $k,n\in
\mathbb N$.
