  We prove a closed formula for leading Gopakumar- Vafa BPS invariants of local
Calabi-Yau geometries given by the canonical line bundles of toric Fano
surfaces. It shares some similar features with Goettsche-Yau-Zaslow formula:
Connection with Hilbert schemes, connection with quasimodular forms, and
quadratic property after suitable transformation. In Part I of this paper we
will present the case of projective plane, more general cases will be presented
in Part II.
