  We propose a Nekrasov-type formula for the instanton partition functions of
four-dimensional N=2 U(2) gauge theories coupled to (A_1,D_{2n})
Argyres-Douglas theories. This is carried out by extending the generalized AGT
correspondence to the case of U(2) gauge group, which requires us to define
irregular states of the direct sum of Virasoro and Heisenberg algebras. Using
our formula, one can evaluate the contribution of the (A_1,D_{2n}) theory at
each fixed point on the U(2) instanton moduli space. As an application, we
evaluate the instanton partition function of the (A_3,A_3) theory to find it in
a peculiar relation to that of SU(2) gauge theory with four fundamental
flavors. From this relation, we read off how the S-duality group acts on the UV
gauge coupling of the (A_3,A_3) theory.
