Complicated and exotic expansions of solutions to the Painlevé equations

We consider the complicated and exotic asymptotic expansions of solutions to a polynomial ordinary differential equation (ODE). They are such series on integral powers of the independent variable, which coefficients are the Laurent series on decreasing powers either of the logarithm of the independent variable or on its pure imaginary power correspondingly. We propose an algorithm for writing ODEs for these coefficients. The first coefficient is a solution of a truncated equation. For some initial equations, it is a polynomial. Question: will the following coefficients be polynomials? Here the question is considered for the third ($P_3$), fifth ($P_5$) and sixth ($P_6$) Painlevé equations. We have found that second coefficients in seven of eight families of complicated expansions are polynomials, as well in two of four families of exotic expansions, but in other three families, polynomiality of the second coefficient demands some conditions. We give detailed proofs and calculations of these results.