Sum of binomials when only the sum is observed

Thanks for the reply. Since I assume the p-s are quite different, I couldn’t find any simple analytical solution. The most relevant approximation I found was this R package:
https://cran.r-project.org/web/packages/sinib/index.html
Which is based on this paper:


Not really sure what to do with it, though… I assumed integrating an external package within the Stan code isn’t possible.