Are admissions offers sent after the April 15 deadline?
Yes, sometimes.

(I am the Graduate Coordinator of the UGA math department, i.e., the faculty member who is most directly in charge of graduate admissions here.)  April 15th is "the singularity" for sure: we make offers months before, many of which are held until the weeks, days or hours before April 15th.  As the deadline approaches, we try to check back in with people and try to hear sooner rather than later if they will decline.

While we try our best to get everything resolved by April 15th, it is likely that we will not entirely succeed.  E.g. for us this year we made an offer that was received (just) after April 15th.  Things tend to move very quickly after April 15th as well: we can really burn through a waiting list when it turns out that a lot of people on it took something by the April 15th deadline.

If you are still interested in a department and haven't been rejected yet by them, I would say that today is a great today to write back and let them know you're still interested.  At this time of year, knowing that an offer is likely to be accepted is worth something to us.

Good luck.