(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

The $N\times N$ determinant $$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$ has the nice form $$D(a,\vec{b})=\prod_{j=1}^N(N+a+b_j-j)!\prod_{i=j+1}^N(b_j-b_i-j+i).$$

I would like to know if the generalization where $a$ is allowed to vary with $i$ has a nice expression as well, $$D(\vec{a},\vec{b})=\det\left((2N+a_i+b_j-i-j)!\right)=?$$

I know Krattenthaler has this great paper about determinants, but I was not able to find help there.