  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.
