  Let S be a nonsingular projective surface. Each vector bundle V on S of rank
s induces a tautological vector bundle over the Hilbert scheme of n points of
S. When s=1, the top Segre classes of the tautological bundles are given by a
recently proven formula conjectured in 1999 by M. Lehn. We calculate here the
Segre classes of tautological bundles for all ranks s over all K-trivial
surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all
surfaces, thus establishing a full analogue of Lehn's formula. We also give
conjectural formulas for certain series of Verlinde Euler characteristics over
the Hilbert schemes of points.
