What is the inverse of the following function? What is the inverse function of $ y=x^3+x$? thanks
According to Wolfram Mathematica, the equation $x^3+x=y$ is solved by
$$
x=\frac{\sqrt[3]{\sqrt{3} \sqrt{27 y^2+4}+9 y}}{\sqrt[3]{2}
   3^{2/3}}-\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 y^2+4}+9 y}}.
$$
I get $x= y-y^3+3y^5-12y^7+55y^9-273y^{11}+-...$ where the coefficients are given by http://oeis.org/A001764 with alternating signs.
This is of course convergent up to the point where the closed form solution has its poles. The reference shows that, as a generating function, the coefficients have nice closed forms in terms of binomial coefficients.