Find the first coefficients of the inverse series of $x+x^2\sqrt{1+x}$

Question

This is exercise 2c from chapter 2 of Wilf's generatingfunctionology. The problem is to find the inverse series $g(x)$ of the series of $f(x)=x+x^2\sqrt{1+x}$ (i.e. $f(g(x))=g(f(x))=x$). I get nowhere with the solution he gives in the back of the book (to substitute the inverse series $g(x)$ into $f(x)$). This was easy to do for sine and tangent but the method given earlier in the chapter doesn't work (as far as I can tell) with the square root.

Answer 1

You know $f(0) = 0$, so you want $g(0) = 0$. Note that $f(t) = t + t^2 (1 + t/2 + \ldots) = t + t^2 + t^3/2 + \ldots$. Plug in $t = g(x) = a_1 x + a_2 x_2 + a_3 x^3 + \ldots$.