In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman diagrams and Deligne Mumford space.")
The expression in parentheses in the table can be identified with OEIS A263634, the logarithmic polynomials with important applications to the operator calculus of Appell sequences, to moment-cumulant theory, to isolating indeterminates in the construction of power series compositions, and to the related algebra of symmetric polynomials/functions as an avatar (e.g.f. vs. o.g.f. rep) of the Faber polynomials for transforming the symmetric power sum polynomials into either the complete symmetric polynomials or the elementry symmetric polynomials, i.e., the Newton-Waring identities.
The partition polynomials preceding the logarithmic polynomials remain relatively mysterious to me. I would like to check if these polynomials or a reduced form of them are already in the OEIS. Does anyone have an extension to higher orders of this part of the table, or can someone present a simple method of extending it?
(The table in the upper part of the page contains essentially the first few partition polynomials for Lagrange compositional inversion of formal Taylor series / e.g.f.s, OEIS A134685).