Name of the probability distribution If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows:
$$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$
I'm interested in the properties of this distribution. I could derive them, but maybe it has some special name, so I could search for it specifically.
So, my question, what is the name of the above distribution with the above pdf?

An explicit derivation of Robert Israel's claim is fairly straightforward by induction:  $$\begin{align*}  \operatorname{E}[X^{2m}] &= \int_{x=-\infty}^\infty x^{2m} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx \end{align*}$$ and with the choice $$u = x^{2m-1}, \quad du = (2m-1) x^{2(m-1)} \, dx, \quad dv = x e^{-x^2/2} \, dx, \quad v = -e^{-x^2/2},$$ we obtain $$\begin{align*} \operatorname{E}[X^{2m}] &= \left[ -\frac{x^{2m-1} e^{-x^2/2}}{\sqrt{2\pi}} \right]_{x=-\infty}^\infty + (2m-1)\int_{x=-\infty}^\infty x^{2(m-1)} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx \\ &= (2m-1)\operatorname{E}[X^{2(m-1)}]. \end{align*}$$  Since $\operatorname{E}[X^0] = 1$, we obtain $$\operatorname{E}[X^{2m}] = \prod_{j=1}^m (2j-1) = (2m-1)!! = \frac{(2m)!}{2^m m!}.$$
