  In this paper we obtain an explicit formula for the number of curves in two
dimensional complex projective space, of degree d, passing through
d(d+3)/2-(k+1) generic points and having one node and one codimension k
singularity, where k is at most 6. Our main tool is a classical fact from
differential topology: the number of zeros of a generic smooth section of a
vector bundle V over M, counted with a sign, is the Euler class of V evaluated
on the fundamental class of M.
