In the coinductive calculus of streams (sensu Rutten) $\exp(rX) = 1/(1-rX)$.
Is there a similarly nice representation for $\exp(rX^2)$?
Edit: I've just received a downvote on this. I'm making a wild guess this is because of the weakness of the connection to category-theory, so I've removed the tag. It could also be that this is opaque if you don't know what stream calculus is. Not sure what to do about that. If you do downvote a comment would be helpful.
Edit 2: Maybe this will help: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.873.6588&rep=rep1&type=pdf