  We study 4d type ${\cal H}_0$ Argyres-Douglas theory in $\Omega$-background
by constructing Liouville irregular state of rank 5/2. The results are compared
with generalized Holomorphic anomaly approach, which provides order by order
expansion in $\Omega$-background parameters $\epsilon_{1,2}$. Another crucial
test of our results provides comparison with respect to Painlev\'{e} 1
$\tau$-function, which was expected to be hold in self-dual case
$\epsilon_1=-\epsilon_2$. We also discuss Nekrasov-Shatashvili limit
$\epsilon_1=0$, accessible either by means of deformed Seiberg-Witten curve, or
WKB methods.
