Cubic series reversion ($g=f/a_1-a_2{f}^2/a_1^3+(2a_2^2-a_1{a}_3)f^3/a_1^5+O(f^4)$)

Suppose $f=a_{1}g+a_2{g}^2+a_3{g}^3+O(g^4)$ with $a_1\ne 0$. Substitute $g=b_{1}f+b_2{f}^2+b_3{f}^3+O(f^4)$ and compare coefficients. The three equations are $a_1{b}_1=1$, $a_1{b}_2+a_2{b}_1^2=0$, and $a_1{b}_3+2a_2{b}_1{b}_2+a_3{b}_1^3=0$. Solving them proves the title formula. This is series reversion for a local inverse function; analyticity guarantees a convergent inverse series sufficiently near the origin.