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We study 4d type ${\cal H}_0$ Argyres-Douglas theory in $\Omega$-backgroundby constructing Liouville irregular state of rank 5/2. The results are comparedwith generalized Holomorphic anomaly approach, which provides order by orderexpansion in $\Omega$-background parameters $\epsilon_{1,2}$. Another crucialtest 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, orWKB methods.

A note on rank 5/2 Liouville irregular block, Painlev\'e 1 and the ${\cal H}_0$ Argyres-Douglas theory