Let $K_\alpha$ denote the modified Bessel function of the second kind and order $\alpha$. We have
$$ K_{1/2}(1) = 2e\sqrt{\pi /2},\\ K_{3/2}(1) = 7e\sqrt{\pi /2},\\ K_{5/2}(1) = 37e\sqrt{\pi /2},\\ K_{7/2}(1) = 266e\sqrt{\pi /2},\\ \text{etc.} $$
It seems like we always get an integer multiple of $e\sqrt{\pi /2}$. Does there exist a closed-form or otherwise interesting expression for the integer prefactor? How about other types of Bessel functions?