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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 $$\frac{1}{1}-\frac{1}{3}+\frac{1}{10}-\frac{1}{35}+\frac{1} {126}-\frac{1}{462}+\frac{1}{1716}-\frac{1}{6435}+\frac{1}{24310}-\frac{1}{92378} +\frac{1}{352716} =0.744327739.$$

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up vote 1 down vote accepted

Mathematica can do it: The sum is

$$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{2n+1}{n+1}} = \frac{2}{25}(5 + 4\sqrt{5}\,\operatorname{csch}^{-1}(2)).$$

Wolfram alpha computation

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