We compute certain 2K+4-point, one-loop couplings in the type IIA string
compactified on K3 x T^2, which are related to a topological index on this
manifold. Their special feature is that they are sensitive to only short and
intermediate BPS multiplets. The couplings derive from underlying prepotentials
G[K](T,U), which can be nicely summed up into a fundamental generating
function. In the dual heterotic string on T^6, the amplitudes describe
non-perturbative gravitational corrections to K-loop amplitudes due to bound
states of fivebrane instantons with heterotic world-sheet instantons. We argue,
as a consequence, that our results also give information about instanton
configurations in six dimensional Sp(2k) gauge theories on T^6.Comment: 32 p, harvmac, 1 fig. Revision: taking the fermionic contractions
into account, the K3 elliptic genus disappear
