Find coefficient in series expansion of $x^2C^3(x)$

Question

I need to find the $n$-th coefficient of the expression $P(x)=x^2C^3(x)$ where $C(x)$ is the generating function for the Catalan numbers: $$C(x)=\frac{1-\sqrt{1-4x}}{2x}$$

I'm trying to solve the problem via Lagrange inversion formula, using this fact, $$[x^n]P(x)=[x^{n-2}]C^3(x),$$ and $C(x)$ satisfies the functional equation $$C(x)=1+xC^2(x),$$

do you know how to use this equation for the formula?

Answer 1

$xC^2(x)=C(x)-1$ gives $$P(x)=xC(x)(C(x)-1)=xC^2(x)-xC(x)=C(x)-1-xC(x),$$ hence $P_0=0$ and $$\forall n\ge1\quad P_n=C_n-C_{n-1}.$$