Why is $\frac{1}{m+1} = \sum_{n=0}^\infty \frac{2^{-2n-m-1} (2n+m)!}{(n+m+1)!n!}$? I have the strong assumption that for $m \in \mathbb N_0$ we have
$$\frac{1}{m+1} = \sum_{n=0}^\infty \frac{2^{-2n-m-1} (2n+m)!}{(n+m+1)!n!}=\sum_{n=0}^\infty \frac{2^{-2n-m-1}}{2n+m+1}\binom{2n+m+1}{n}.$$
But I didn't find an explicit way to show it.
In case you wanna know where this series is from: I'm still trying to solve How to prove an equality involving an infinity series and the modified Bessel functions?. If you plug in $s=\alpha=\beta=1$ you eventually come up with the series equation from above. At least the series equation would show the special case (in fact the other direction is not true).

I guess I post my approch which is based on the idea of Claude Leibovici for clarity.
Using Legendre duplication we have
$$(2n+m)!=\Gamma(2n+m+1) = \Gamma(n+ \tfrac{m+1}{2})\Gamma(n+ \tfrac{m+2}{2})2^{2n+m}/\sqrt{\pi}.$$
So we have
$$\sum_{n=0}^\infty \frac{2^{-2n-m-1} (2n+m)!}{(n+m+1)!n!}=\frac{1}{2\sqrt{\pi}} \sum_{n=0}^\infty \frac{ \Gamma(n+ \tfrac{m+1}{2})\Gamma(n+ \tfrac{m+2}{2})}{\Gamma(n+m+2)n!}\\
= \frac{1}{2\sqrt{\pi}} \frac{ \Gamma(\tfrac{m+1}{2})\Gamma(\tfrac{m+2}{2})}{\Gamma(m+2)} F(\tfrac{m+1}{2},\tfrac{m+2}{2},m+2,1)\\
\frac{1}{2\sqrt{\pi}} \frac{ \Gamma(\tfrac{m+1}{2})\Gamma(\tfrac{m+2}{2})}{\Gamma(m+2)} \frac{\Gamma(m+2)\Gamma(m+2-\tfrac{m+1}{2}-\tfrac{m+2}{2})}{\Gamma(m+2-\tfrac{m+1}{2})\Gamma(m+2-\tfrac{m+2}{2})}\\
= \frac{1}{2\sqrt{\pi}} \frac{ \Gamma(\tfrac{m+1}{2})\Gamma(\tfrac{m+2}{2})}{\Gamma(m+2)} F(\tfrac{m+1}{2},\tfrac{m+2}{2},m+2,1)\\
\frac{1}{2\sqrt{\pi}} \frac{ \Gamma(\tfrac{m+1}{2})\Gamma(\tfrac{m+2}{2})}{\Gamma(m+2)} \frac{\Gamma(m+2)\Gamma(\tfrac{1}{2})}{\Gamma(\tfrac{m}{2}+\tfrac{3}{2})\Gamma(\tfrac{m}{2}+\tfrac{2}{2})}
=\frac{1}{m-1}.
$$
