Calculation of complicated asymptotic expansions of solutions to the Painlevé equations

We consider the complicated 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 of the logarithm of the independent variable. 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 ($P_3$) and sixth ($P_6$) Painlevé equations. It appears that for them the second coefficients are polynomials in all cases, but the third coefficient is a polynomial ether always, either under some restriction on parameters, or never.