Neumann polynomial

The Neumann polynomials are defined by the recurrence $$\left\{ \begin{array}{ll} O_0(s)&=\dfrac{1}{s} \\ O_1(s)&=\dfrac{1}{s^2}\\ O_{n}(s)&=O_{n-2}(s)-2O_{n-1}'(s);n \geq 2 \end{array} \right.$$ They also obey the explicit formula $$O_n(s) = \dfrac{n}{4} \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(n-1-k)!(\frac{2}{s})^{n+1-2k}}{k!}; n \geq 1.$$