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$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

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

See also pages 33 and 34 of "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" 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 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.)

$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

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

See also pages 33 and 34 of "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" 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 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.)

$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

(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 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.)

$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

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

See also pages 33 and 34 of "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" 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 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.)

$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

(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 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.)

$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 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), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" 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".

(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 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.)