I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau threefolds, and their modularity. Given a specific CY 3-fold and a fixed K3 lattice polarization, how can one systematically compute the DT invariants. What data of the CY 3-fold do I need? I am interested in not just smooth CYs, but smooth ones is a natural starting point, and I don't know how to handle those.

Are there any references, where such computations have been worked out, maybe in very simple cases.