IR-MO 2014 Problem 8

The polynomials $k_n(x_1, \ldots, x_n)$ , where $n$ is a non-negative integer, satisfy the following conditions $$k_0=1$$ $$k_1(x_1)=x_1$$ $$k_n(x_1, \ldots, x_n) = x_nk_{n-1}(x_1, \ldots , x_{n-1}) + (x_n^2+x_{n-1}^2)k_{n-2}(x_1,\ldots,x_{n-2})$$ Prove that for each non-negative $n$ we have $k_n(x_1,\ldots,x_n)=k_n(x_n,\ldots,x_1)$ .