  BPS invariants are computed, capturing topological invariants of moduli
spaces of semi-stable sheaves on rational surfaces. For a suitable stability
condition, it is proposed that the generating function of BPS invariants of a
Hirzebruch surface takes the form of a product formula. BPS invariants for
other stability conditions and other rational surfaces are obtained using
Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are
given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The
applied techniques can be applied iteratively to compute invariants for higher
rank.
