  In this article we provide another method for obtaining explicit formulas
yielding counts of secant planes to a projective curve. We formulate the
problem in terms of Segre classes of suitable bundles over the symmetric
product of the curve and take advantage of the result of Ellingsrud,
G\"ottsche, and Lehn concerning the structure of integrals of polynomials in
Chern classes of tautological bundles over Hilbert schemes of points. We use
this to set up a recursion starting from the easy to compute case of the
projective line.
