How to simplify $P_d(n,k)=H_d(n,k)/k!$ with $H_d(n,k)$ defined as
$$H_d(n,k+1) = k! \sum_{j=0}^k \frac{ (-1)^j }{(k-j)!} \sum H_d(n/d, k-j)$$
where the inside sum is taken over all $d$ such that $d|n$ and for $d\geq 2$, is a $(j+1)$-st power ?
I have no idea how to deal with the recursion...
What real tools would you employ for calculating the integral below?
$$\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi \sqrt{1-u}}{u-2}+\pi } }{u\sqrt{1-u}} \, du$$
