The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero

Let $\xi_1, \ldots, \xi_n$ be independent random variables with $\mathbf{E}\xi_i=0,$ $\mathbf{E}|\xi_i|^t<\infty$, $t>2$, $i=1,\ldots, n,$ and let $S_n=\sum_{i=1}^n \xi_i.$ In the present paper we prove that the exact constant ${\overline C}(2m)$ in the Rosenthal inequality
$$ \mathbf{E}|S_n|^t\le C(t) \max \Bigg(\sum_{i=1}^n\mathbf{E}|\xi_i|^t,\ \Bigg(\sum_{i=1}^n \mathbf{E}\xi_i^2\Bigg)^{t/2}\Bigg) $$
for $t=2m,$ $m\in \mathbf{N},$ is given by
$$ \overline C(2m)=(2m)! \sum_{j=1}^{2m} \sum_{r=1}^j \sum \prod_{k=1}^r \frac {(m_k!)^{-j_k}} {j_k!}, $$
where the inner sum is taken over all natural $m_1 > m_2 > \cdots > m_r > 1$ and $j_1, \ldots, j_r$ satisfying the conditions $m_1j_1+\cdots+m_rj_r=2m$ and $j_1+\cdots+j_r=j$. Moreover
$$ \overline C(2m)=\mathbf{E}(\theta-1)^{2m}, $$
where $\theta $ is a Poisson random variable with parameter 1.