Complicated and exotic expansions of solutions to the fifth Painlevé equation

For the fifth Painlevé equation $(P_5)$, we calculate the second coefficients of the complicated and exotic asymptotic expansions of its solutions. The equation $(P_5)$ has two different cases I and II giving such expansions. Two complicated expansions (main and additional) and one exotic exist in each case. It appears that the second coefficients in $3$ complicated expansions are polynomials, but for the main expansion in the case I, one condition on parameters is necessary and sufficient for that. The exotic expansions are always Laurent polynomials only in the case I, and in the case II, two conditions on parameters are necessary and sufficient for that.