  We give an algorithmic description of the action of the Chern classes of
tautological bundles on the cohomology of Hilbert schemes of points on surfaces
within the framework of Nakajima's oscillator algebra. This leads to an
identification of the cohomology ring of Hilbert schemes of the affine plane
with a ring of differential operators on a Fock space. We end with the
computation of the top Segre classes of tautological bundles associated to line
bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
