  We establish an explicit formula for the character of an irreducible
finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a
finite sum with integer coefficients in terms of a basis $\mathcal{E}_{\mu}$
(Euler characters) of the character ring. We prove a simple formula for the
behaviour of the ``superversion'' of $\mathcal{E}_{\mu}$ in the
$\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$-case under the map $ds$ on the
supercharacter ring induced by the Duflo-Serganova cohomology functor $DS$. As
an application we get combinatorial formulas for superdimensions, dimensions
and $\mathfrak{g}_0$-decompositions for $\mathfrak{gl}(m|n)$ and
$\mathfrak{osp}(m|2n)$.
