Enumeration of n-fold tangent hyperplanes to a surface

For each 1≤ n≤6 we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in {n} by a complete system of curves of genus n as well as the number {17,601,000} of rational ({ singular}) plane quintic curves in a generic quintic threefold.