  We consider the half-genus expansion of the resolvent function in the
$\beta$-deformed matrix model with three-Penner potential under the AGT
conjecture and the $0d-4d$ dictionary. The partition function of the model,
after the specification of the paths, becomes the DF conformal block for fixed
$c$ and provides the Nekrasov partition function expanded both in $g_s =
\sqrt{-\epsilon_1 \epsilon_2}$ and in $\epsilon = \epsilon_1+\epsilon_2$.
Exploiting the explicit expressions for the lower terms of the free energy
extracted from the above expansion, we derive the first few $\epsilon$
corrections to the Seiberg-Witten prepotential in terms of the parameters of
SU(2), $N_{f} =4$, ${\cal N}= 2$ supersymmetric gauge theory.
