$F_t(x) = (1-tx)^{-1/t} - 1$ is the e.g.f. for enumerating plane $m$-ary trees when $t=m-1$. OEIS [A094638][1] provides some examples when $t = \pm 1,\pm2,\pm3$ in my Dec. 15, 2007, comments. The e.g.f. is of importance for $t$ any real number.

For $t = 3$, the e.g.f. is $F_3(x) = (1-3x)^{-1/3} - 1 = x + 4 x^2/2! + 28 x^3/3! + 280 x^4/4! + \cdots$, generating the sequence [A007559][2], the right triple factorials, enumerating the number of increasing quaternary trees on n vertices.

For $t = -3$, the e.g.f. is $F_{-3}(x) = (1+3x)^{1/3}-1 = -x - 2x^2/2! + 10 x^3/3! - 80 x^4/4! + \cdots$, generating the signed sequence [A008544][3], the left triple factorials, enumerating increasing plane (a.k.a. ordered) trees with n vertices (one of them a root labeled 1) where each vertex with outdegree r >= 0 comes in r+1 types (like an (r+1)-ary vertex).

Note the relation to multiplicative inversion: $1+F_t(x) = 1/(1+F_{-t}(-x))$. (I called this inversion, [A133314][4], the list partition transform, in the A094638 for historical reasons in my journey of discovery of its significance.)  

The compositional inverse (CI) plays a role also. The CI of $G(x)$ about the origin of $F(x)$ is $G(x,t)= [1-(1+x)^{-t}]/t$, so the infinitesimal generator / Lie vector for generating $F(x)$ is $g(x)\partial_x = \frac{1}{G'(x,t)}\partial_x = (1+x)^{t+1}\partial_x$; that is, $\exp[x g(z)\partial_z] \; z |_{z=0}= F(x)$. The Cayley analytic trees associated with the iterated operation $(g(z)\partial_z)^n \; z |_{z=0}$ for $t=\pm3$ are of the types described above (see. e.g., my "Mathemagical Forests" reffed in [A145271][5] along with the Bergeron et al. ref "Varieties of Trees" and my "Addendum to Mathemagical Forests" in A094638). (This is all related to the formalism of pre-Lie algebras.) 

My comments on the relation between $[A^{(m)}]$ for $m \geq 1$ and the Fuss-Catalan numbers, generated by compositional inversion of $f(x) = x \pm x^{m+1}$ about $x=0$ (see, e.g., [A001764][6]), in my answer / extension to the MO-Q "[Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution][7]" presents another perspective since $m$ is extended there to any integer. This is a generalization of the formalism of Novelli and Thibon in "[Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions][8]". 

On Fuss-Catalan numbers and relations among their generating functions, see also pages 33 and 34 of "[Connecting Scalar Amplitudes using The Positive Tropical Grassmannian][11]" by Cachazo and Umbert.       

My formulas in OEIS A094638, as pointed out therein and above, involve both compositional inversion (in fact $[A^{(1)}]$ for CI is normalized [A133437][9]) and multiplicative inversion ([A133314][4]) and so are naturally related to Koszul duality as noted in the MO-Q "[Inversion, Koszul duality, combinatorics and geometry][10]". In fact, as I remarked in the formula section of A094638 and repeat above: With $F(x,t) = (1-tx)^{-1/t} - 1$ an e.g.f. for the row polynomials $P(n,t)$ of A094638 with $P(0,t)=0$, $G(x,t)= [1-(1+x)^{-t}]/t$ is the CI in $x$. The case for $t=3$ is used in the proof of Corollary 4.2. of "Associator dependent algebras and Koszul duality" by Bremner and Dotsenko.

(Often the presence of negative integers in a generating function indicates some combinatorics of an underlying topological nature, such as Euler's formula for polytopes. This [MO-Q][12] contains another example of how natural it can be to extend $n$ in significant combinatorial sequences from the natural numbers to the full integers and retain combinatorial import.) 

  


  [1]: https://oeis.org/A094638
  [2]: https://oeis.org/A007559
  [3]: https://oeis.org/A008544
  [4]: https://oeis.org/A133314
  [5]: https://oeis.org/A145271
  [6]: https://oeis.org/A001764
  [7]: https://mathoverflow.net/questions/422539/infinite-dimensional-involutions-infinitely-large-sets-of-multivariate-polynomi
  [8]: https://arxiv.org/abs/1403.5962
  [9]: https://oeis.org/A133437
  [10]: https://mathoverflow.net/questions/194888/inversion-koszul-duality-combinatorics-and-geometry
  [11]: https://arxiv.org/abs/2205.02722
  [12]: https://mathoverflow.net/questions/412573/combinatorics-for-the-action-of-virasoro-kac-schwarz-operators-partition-poly