  Let X be the total space of the canonical bundle of P^2. We study the
generalized Donaldson-Thomas invariants, defined in the work of Joyce-Song, of
the moduli spaces of the 2-dimensional Gieseker semistable sheaves on X with
first Chern class equal to k times the class of the zero section of X. When
k=1, 2 or 3, and semistability implies stability, we express the invariants in
terms of known modular forms. We prove a combinatorial formula for the
invariants when k=2 in the presence of the strictly semistable sheaves, and
verify the BPS integrality conjecture of Joyce-Song in some cases.
