$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS [A094638][1] provides some examples when $t = -1,-2,-3$  in my Dec. 15, 2007, comments. The e.g.f. is of importance for $t$ any real number.

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][2]), in my answer / extension to the MO-Q "[Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution][3]" has 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][3]".

My formulas in OEIS A094638, as pointed out therein, involve both compositional inversion (in fact $[A^{(1)}]$ is normalized [A133437][5]) and multiplicative inversion ([A133314][6]) and so are naturally related to Koszul duality as noted in the MO-Q "[Inversion, Koszul duality, combinatorics and geometry][7]".

See also pages 33 and 34 of "[Connecting Scalar Amplitudes using The Positive Tropical Grassmannian][8]" by Cachazo and Umbert.

(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][4] 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/A001764
  [3]: https://arxiv.org/abs/1403.5962
  [4]: https://mathoverflow.net/questions/412573/combinatorics-for-the-action-of-virasoro-kac-schwarz-operators-partition-poly
  [5]: https://oeis.org/A133437
  [6]: https://oeis.org/A133314
  [7]: https://mathoverflow.net/questions/194888/inversion-koszul-duality-combinatorics-and-geometry
  [8]: https://arxiv.org/abs/2205.02722