Revamped extended comment, 3/20/23: The initial eqn. $$T_n=1+xT_n^n$$ is eqn. 6.3 on p. 449 of "Functional composition patterns and power series reversion" by Raney. As you state in your comment, for $n= 3$, the solution is the $(2)$-Fuss-Catalan sequence of numbers OEIS [A001764][1] = (1,1,3,12,55,...) . These are the absolute values of the coefficients of the top order monomials $u_1^n$, the diagonals, of the set of $(2)$-associahedra partition polynomials $A^{(2)}_n(u_1,u_2...,u_n)$ that are the coefficients of the compositional inverse $$(O^{(2)}(z))^{(-1)} = z + A_1(u_1)z^{2 \cdot 1+1} + A_2(u_1,u_2) u_2 z^{2\cdot 2 +1} + A_3(u_1,u_2,u_3) u_3 z^{2\cdot 3 +1} + \cdots.$$ of the odd o.g.f. $$O^{(2)}(z) = z + u_1z^{2 \cdot 1+1} + u_2 z^{2\cdot 2 +1} + u_3 z^{2\cdot 3 +1} + \cdots.$$ The absolutes values of the sum of the signed coefficients of the polynomials of $[A^{(2)}]$ give [A000108][2] = (1,1,2,5,14, ...), the Catalans. For $m$ any integer, there is a ladder of sets of $(m)$-noncrossing partitions / $(m)$-Narayana polynomials, $[N^{(m)} = [N]^m$, dual to the $(m)$-associahedra polynomials, $[A^{(m)}]$. The duals are characterized by the generalized face-h-polynomial substitution identity / raising-lowering identity $$[A^{(m)}] = [N^{(m)}][A^{(0)}] = [N^{(m)}][R] = [N]^m [R],$$ or, conversely, $$[A^{(m)}][R] = [N^{(m)}] = [N]^m ,$$ where $[A^{(0)}] =[R]$ is the set of reciprocal partition polynomials that can be defined by the shifted reciprocal $$\frac{x}{O(x)} = \frac{1}{1+u_1x +u_2 x^2 + u_3 x^3+\cdots} = \sum_{n \geq 0} R_n(u_1,...,u_n) x^n.$$ This f-h identity implies that summing the coefficients of one set gives the diagonals of the other set. And, indeed, summing the coefficients of the dual set $[N^{(2)}]$ generates the $(2)$-Fuss-Catalan sequence and the diagonal is [A000108][2]. The set $[A^{(2)}]$ can be generated by the Lagrange inversion formula; e.g., using Wolfram Alpha online, $A^{(2)}_4 = −u_4 + 10 u_1 u_3 + 5 u_2^2 − 55 u_1^2 u_2 + 55 u_1^4 $ is generated by eighth derivative (1/(2 \cdot 4)!) (1 + u_1x^2 + u_2x^4 + u_3x^6 + u_4x^(2 \cdot 4))^(-(2 \cdot 4+1)) / (2 \cdot 4+1) at x = 0 and $N^{(2)}_4 = 14 u_1^4 + 28 u_2 u_1^2 + 8 u_3 u_1 + 4 u_2^2 + u_4$ is generated by eighth derivative (1/(2 \cdot 4)!) (1 + u_1x^2 + u_2x^4 + u_3x^6 + u_4x^(2 \cdot 4))^((2 \cdot 4+1)) / (2 \cdot 4+1) at x = 0. For $m=-2$, simply replace in an obvious fashion the $2$ in the formulas above with $-2$ to obtain (with compositional inversion away from the origin, i.e., about the the point at infinity) the $(-2)$-Fuss-Catalan sequence of numbers (1,A000108) =(1,1,1,2,5,14,...); e.g., a generalized Lagrange inversion formula gives $A^{(-2)}_4 = - (5 u_1^4 + 15 u_2 u_1^2 + 6 u_3 u_1 + 3 u_2^2 + u_4)$ is generated by eighth derivative (1/(2 \cdot 4)!) (1 + u_1x^2 + u_2x^4 + u_3x^6 + u_4x^(2 \cdot 4))^(-(-2 \cdot 4+1)) / (-2 \cdot 4+1)) at x = 0 and $N^{(-2)}_4 = -30 u_1^4 + 36 u_2 u_1^2 - 8 u_3 u_1 - 4 u_2^2 + u_4$ is generated by eighth derivative (1/(2 \cdot 4)!) (1 + u_1x^2 + u_2x^4 + u_3x^6 + u_4x^(2 \cdot 4))^(-2 \cdot 4+1)) / (-2 \cdot 4+1) at x = 0. The absolutes values of the sum of the signed coefficients of the polynomials of $[A^{(-2)}]$ give (1, [A006013][3]) = (1,1,2,7,30,143, ...). The absolute values of the sum of the signed coefficients of the polynomials of $[N]^{-2}$ give (1,[A000108][2]) = (1,1,1,2,5,14,...), the shifted Catalan numbers. The absolutes of the diagonals of $[N]^{-2}$ give (1, A006013). For $m=-3$, the absolutes of the diagonals of $[A^{(-3)}]$ are (1,[A006013][3]) =(1,1,2,7,30,143,...), the $(-3)$-Fuss-Catalan sequence of numbers. The absolute sums of the coefficients of $A^{(-3)}_n(u_1,...,u_n)$ give the $(-3)$-Fuss-Narayana sequence $(1,[A006632][5]) = (1,1,3,15,91,612, ...)$ as do the 'diagonals' of $[N^{(-3)}]$ whereas the absolutes of the sums of $[N^{(-3)}]$ are (1, A006013). For any integer $m$, the reduced polynomials of $[A^{(m)}]$ are the coefficients of the compositional inverse about $x=0$ of $(RTA^{(m)}(x,t))^{(-1)} = \frac{x}{(1+(1+t)x)(1+x)^{m}}$ or $(TA^{(m)}(x,t))^{(-1)} = \frac{x}{(1+(1+t)x)(1+tx)^{m}} .$ **Examples**: $[A^{(2)}]$, reduced is [A243662][4] and [A102537][5] $A^{(2)}_3 = -u_3 + 8 u_1 u_2 - 12 u_1^3$ second derivative [(1+(1+t)x)(1+x)^2]^3 / 3! at x = 0 is 12 + 8t + t^3 second derivative [(1+(1+t)x)(1+tx)^2]^3 / 3! at x = 0 is 1 + 8t + 12t^2. $[A^{(-2)}]$, reduced is [A286784][6] $A^{(-2)}_3 = - (2 u_1^3 + 4 u_2 u_1 + u_3) $ second derivative [(1+(1+t)x)(1+x)^(-2)]^3 / 3! at x = 0 is t^2 - 4 t + 2 second derivative [(1+(1+t)x)(1+tx)^(-2)]^3 / 3! at x = 0 is 2 t^2 - 4 t + 1. The generalized f-h identity inplies $(RTN^{(m)}(x,t))^{(-1)} = \frac{x}{(1+tx)(1+x)^{m}} $ so $(TN^{(m)}(x,t))^{(-1)} = \frac{x}{(1+x)(1+tx)^{m}}.$ **Examples**: $[N]^2$, reduced is [A120986][7] and [A108767][8] $N^{(2)}_3 = u_3 + 6 u_1 u_2 + 5 u_1^3$ second derivative [(1+tx)(1+x)^2]^3 / 3! at x = 0 is 5 + 6 t + t^2 second derivative [(1+x)(1+tx)^2]^3 / 3! at x = 0 is 1 + 6 t + 5 t^2 $N^{(-2)}_3 = 7 t^3 - 6 t^2 + t$ $ second derivative [(1+tx)(1+x)^(-2)]^3 / 3! at x = 0 is t^2 - 6 t + 7 second derivative [(1+x)(1+tx)^(-2)]^3 / 3! at x = 0 is 7 t^2 - 6 t + 1 The coefficients of these polynomials are gleaned from the formulas on page 15 of "[On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements][9]" by Athanasiadis and Tzanaki. At every level of refinement or coarseness--from the coefficients of the partition polynomials to those of their reductions--the analytic formulas remain immutable. For a leap from positive to negative integers for $m$, the iconic rising-to-falling factorials polynomial identity $$n! \binom{-q}{n} =(-1)^n n! \binom{q-1+n}{n}$$ account for apparent variation in the formulas--as above, so below. Drake addresse this in the section 1.10 Numerator polynomials beginning on p. 58 of [his thesis][10] "An inversion theorem for labeled trees and some limits of areas under lattice paths". The combinatorial interpretations of the $(m)$-Fuss-Catalan and the $(m)$-Fuss-Narayana sequences of numbers are then open to the broader ones of the more refined characters, the $(m)$-associahedra and $(m)$-noncrossing partitions and their polynomial reductions. Athanasiadis and Tzanaki (link above) give some geometric interpretations for the coefficients of the reduced polynomials for $m$ any integer as does Drew Armstrong in [his thesis][11] "Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups" and [Muhle and Tzanaki][12] in "Refined Lattice Path Enumeration and Combinatorial Reciprocity". Song's ref in [A173020][13] should lead to others for $m > 0$. Several articles by Novelli and Thibon (e.g., [one][14] and [two][15]) address $m >0$. The combinatorics are rife with trees, lattice paths, dissected polygons, Feynman diagrams, . . . (see A134264 for $[N]$; A133437. differently normalized $[A]$; A354622, $[N]^2$; and A286784, a reduction of $[A^{(-2)}]$. (See the intro to “[Why Delannoy numbers?][16]” by Bandererier and Schwer on some history of the associated names (they ref Stanley in turn)--in particular, Runyon numbers enumerate Dyck paths and noncrossing partitions. [1]: https://oeis.org/A001764 [2]: https://oeis.org/A000108 [3]: https://oeis.org/A006013 [4]: https://oeis.org/A243662 [5]: https://oeis.org/A102537 [6]: https://oeis.org/A286784 [7]: https://oeis.org/A120986 [8]: https://oeis.org/A108767 [9]: https://arxiv.org/abs/math/0605685 [10]: https://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf [11]: https://arxiv.org/abs/math/0611106 [12]: https://arxiv.org/abs/2207.14544 [13]: https://oeis.org/A173020 [14]: https://arxiv.org/pdf/1403.5962.pdf [15]: https://arxiv.org/abs/2106.08257 [16]: https://arxiv.org/pdf/math/0411128.pdf