how to calculate the asymptotic growth rate of coefficients generating function $T(z)$ satisfied this identity


$T(z)=z+\frac{T(z)^3}{6}+\frac{T(z^2)T(z)}{2}+\frac{T(z^3)}{3}$ 



Asymptotic growth rate of coefficients of generating function
In order to analyze these type of equations one can use the methodology of Flajolet and Sedgewick (Analytic Combinatorics Book). More precisely, from the previous specification one gets that the first coefficients of $T(z)$ are


$$T(z)=z+z^3+z^5+2z^7+4z^9+8z^{11}+17z^{13}+39z^{15}+89z^{17}+211z^{19}$$


Observe now that if $T(z)$ has a finite singularity at $z=\rho<1$ (it cannot be $>1$ as this comes from a combinatorial problem: the sequence of numbers in the expansion is increasing), then both $T(z^2)$ and $T(z^3)$ are analytic functions at $z=\rho$, and hence we can approximate them by its Taylor expansion (which can be deduced from the previous expansion).


Hence, making this approximation gives an equation of the form $T(z)=F(z,T(z))$, where $F$ has positive Taylor coefficients. Under this circumstances, the singular expansion of $T(z)$ around its singularity will be of square-root type, and the position of the singularity will be at the solution of the system of equations


$$u=F(z,u),\, 1=F\_u(z,u),$$


If you compute the solution you get $\rho=0.59597132592884902260$. Then $T(z)$ has a square-root type expansion around this value. You can get then the singular expansion (see Flajolet and Sedgewick), and apply the transference theorem to get that the coefficients grow like $ C n^{-3/2} \rho^{-n}$, with $\rho^{-1}$ roughly $1.6779330757926205637$. To compute the constant $C$ you need the singular expansion + application of the Transference Theorem.


