  We study $SU(2)$ gauge theories coupled to $(A_1,D_N)$ theories with or
without a fundamental hypermultiplet. For even $N$, a formula for the
contribution of $(A_1,D_N)$ to the Nekrasov partition function was recently
obtained by us with Y.~Sugawara and T.~Uetoko. In this paper, we generalize it
to the case of odd $N$ in the classical limit, under the condition that the
relevant couplings and vacuum expectation values of Coulomb branch operators of
$(A_1,D_N)$ are all turned off. We apply our formula to the $(A_2,A_5)$ theory
to find that its prepotential is related to that of the $SU(2)$ gauge theory
with four fundamental flavors by a simple change of variables.
