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