Odd polynomial solutions of the Hermite differential equation ($E=4j+2,j\ge 0$)

For $-W^{\prime \prime }+2x{W}^{\prime }=EW$, the power series coefficients obey $(n+2)(n+1)a_{n+2}=(2n-E)a_n$. The condition $W(0)=0$ makes every even coefficient zero. A nonzero odd series terminates precisely when $E=2d$ with $d$ an odd positive integer. A polynomial of degree $d$ must have this value of $E$ by comparison of leading coefficients, so these are all possibilities. With $W^{\prime }(0)=1$, the first three polynomial solutions are $x$, $x-2x^3/3$, and $x-4x^3/3+4x^5/15$, with $E=2,6,10$. They are scalar multiples of the odd Hermite polynomials.