Well, that second post gave me ONE idea, but I'll admit THIS idea doesn't QUITE add up to a no-start....
Starting up after sitting a while, then running rough & smoothing out, SOUNDS like possibly a "flooded" situation. One cause that's only occasionally seen is an injector that doesn't seal and leaks fuel into the cyl. after shut down.
(there's another similar issue possible I remember seeing here, but I need to check details on that)
You might try the "flooded" starting procedure (foot to the floor to cut fuel injection) next time & see if that helps.