Let's have the following relation $\sum\limits_{1}^k (2m_1-1)(2m_2-1)\cdots(2m_k-1)2^{k-1}/k!$ where $m$ takes all values from 1 to $k$. When $k$ is odd we put a positive sign in front of the obtained integer and when $k$ is even a negative sign.
Does anyone know to express this series in a closed form? Up to $k=11$ we have the following $\frac{1}{1}-\frac{1}{3}+\frac{1}{10}-\frac{1}{35}+\frac{1} {126}-\frac{1}{462}+\frac{1}{1716}-\frac{1}{6435}+\frac{1}{24310}-\frac{1}{92378} +\frac{1}{352716} =0.744327739.$