You might also try following the references and links in A145271 ["Coefficients for expansion of [g(x)d/dx]^n g(x); refined Eulerian numbers for calculating compositional inverse of h(x)= (d/dx)^(-1) 1/g(x)"] and A139605 ["Weights for expansion of (f(x)D_x)^n : coefficients of A-polynomials of Comtet"] of the Online Encyclopedia of Integer Sequences for some ideas. Note that the coefficients of A145271 (your expansion coefficients) are embedded in A139605.
An explicit formula for the coefficients are given in the Comtet reference in A139605 on page 166 as egn. (8) with l=1.
