Recursion relation for instanton counting for $SU(2)$ ${\cal N}=2$ SYM in NS limit of $\Omega$ background

In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter $q$ for ${\cal N}=2$ SYM with up to four antifundamental hypermultiplets in NS limit of $\Omega$ background. We propose a new method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. We also investigate a numerical method for deriving the $A$-cycle period valid for arbitrary values of $q$. Analyzing large $q$ asymptotic we get convincing agreement with an analytic expression deduced from a conjecture by Alexei Zamolodchikov in a different context.