  We compute the conformal anomalies for some higher-derivative (non-unitary)
6d Weyl invariant theories using the heat-kernel expansion in the
background-field method. To this aim we obtain the general expression for the
Seeley-DeWitt coefficient $b_6$ for four-derivative differential operators with
background curved geometry and gauge fields, which was known only in flat space
so far. We consider four-derivative scalars and abelian vectors as well as
three-derivative fermions, confirming the result of the literature obtained via
indirect methods. We generalise the vector case by including the curvature
coupling $FF \mathrm{Weyl}$.
