We compute the equivariant K-theory of torus fixed points of Cherkis bow varieties of affine type A. We deduce formulas for the generating series of the Euler numbers of these varieties and observe their modularity in certain cases. We also obtain refined formulas on the motivic level for a class of bow varieties strictly containing Nakajima quiver varieties. These series hence generalise results of Nakajima-Yoshioka. As a special case, we obtain formulas for certain Zastava spaces. We define a parabolic analogue of Nekrasov's partition function and find an equation relating it to the classical partition function.