With , the Witt vectors when exponentiated give rise to the action
for . (Cf. the MathOverflow question Motivation of the Virasoro algebra.)
The function represents a Lie group under composition with respect to the parameter . That is, with roots always chosen as positive and real for in a suitably small neighborhood of the origin,
and the compositional inverse of is simply
The function is also an exponential generating function for the generalized left and right factorials (for positive and negative integral ) presented in OEIS A094638 (mod sign, index shifts, and an additional initial 1 in some cases).
and the sequence is signed A001147, which I will call the signed right double factorial with an additional initial 1 (cf. A094638). (The numerators and denominators of the reduced fractions are A098597 and A046161, apparently.)
The compositional inverse is
and the sequence is A001147.
We can say the augmented right double factorial as represented in this group is skew invariant under compositional inversion. It is also quasi-invariant under multiplicative inversion with
generating 1,1,-1,3,-15, … .
In contrast, for , we have
generating , signed A007559, the signed, right triple factorial augmented with an initial 1.
The shifted reciprocal gives
generating , the signed left triple factorials A008544 augmented with an initial 1.
Note that the compositional inverse pairs can be related to dual families of trees and also the multiplicative inverse pairs, according to the OEIS entries. See also the MO-Q Combinatorial interpretation of series reversion coefficients and the Gessel link therein, which discusses both multiplicative and compositional inversion. The two types of inversions are also related in general to Hopf algebras/monoids and Koszul duality.
The compositional inverse of can be computed from the coefficients of using the compositional inversion formula A134264 related to free cumulants in free probability theory, non-crossing partitions, and Dyck paths, among other combinatorial constructs. This algorithm allows a transformation of right factorials into left factorials, as does A133437