I need to calculate $E\left[\mathrm{sgn}(X)X^n\right]$, where $\mathrm{sgn}(X)$ denotes the sign of the random variable $X$. I don't know the pdf for $X$, so I can't do the calculation directly. I know the characteristic function, so I have the moments, but I'm not sure if it's possible to handle the sign function.
My first thought was to write $\mathrm{sgn}(X) = \lim_{k\to\infty}\tanh(kX)$, then expand tanh as a Taylor series in $X$, resum, and take the limit. However, resumming the resulting series isn't possible, because the moments are extremely messy.
Is there something obvious I'm missing here, or is my task impossible?