(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
$$D(a,\vec{b})=\prod_{j=1}^N(N+a+b_j-j)!\prod_{i=j+1}^N(b_i+b_j-j+i).$$

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][1] about determinants, but I was not able to find help there.


  [1]: http://emis.ams.org/journals/SLC/wpapers/s42kratt.pdf