Consider an odd number of real numbers $x_1,x_2,\dots x_n$. Let $M$ be their median. What can be said about the Hermite–Fourier expansion of $M=M(x_1,x_2,\dots,x_n)$?   

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. 
The case $n=3$ is already interesting to me. 
In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms? 

Clarification: Note that $M$ is a function of $n$ variables  so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussiamn measure on $\mathbb R^n$.