Series of inverse function

It is pretty obvious that $s\mapsto A(s)$ is odd. If you really only need $a_5$ the simplest thing is to write $$A(s)=s+a_3s^3+a_5s^5+?s^7\ ,$$ whereby the question mark represents a full power series. Then $$A^3(s)=s^3(1+a_3s^2+?s^4)^3$$ and therefore $$0=A(s)+A^3(s)-s=s+a_3s^3+a_5s^5+?s^7+s^3\bigl(1+3(a_3s^2+?s^4)+?s^4\bigr)-s\ .$$ Comparing coefficients gives $a_3=-1$, $\>a_5=3$.