Fractional Captain reported a NMAC during cruise that resulted in a TCAS RA response.

Narrative

We had just leveled off at 4;000 assigned by ATC during a short ferry leg from ZZZ1 to ZZZ when I started to see a target on tcas appear approx. 400' above us. It became closer and at last glance I saw 200' vertical. Initially the tcas ra indicated to climb which seemed counter-intuitive as the aircraft was above us and at that time ATC started to announce a climb and almost simultaneously then it switched to a descent on tcas as well as ATC stated the same. We descended and then heard increase descent on tcas and leveled approx. 3;000' and coordinated with ATC. I don't know if I was able to state on the radios that we were having a Tcas RA because this all happened so quickly.After landing we were able to identify the other aircraft as a Cirrus that departed out of ZZZ and landed back at ZZZ shortly after us. The tail number was Aircraft Y and when looked up belonged to Company X on the field at ZZZ. I called the flight school to try and gain some insight as to what happened but spoke to their Chief Instructor (as he identified himself) and relayed the fact that someone smarter than me had overlayed our flight times and paths and showed we came within 25' of each other. The cirrus shows on their track a maneuver initiated at 7;000' with a linear descent constant to 400'. The person I spoke with has my name and phone number and stated they would call back but I have yet to hear from them. I would appreciate if someone could investigate this a bit further to speak with their pilot or flight school if certain practices being conducted in areas of airspace may not be practical. When I asked an individual inside the fbo we parked at about this Company X they did not seem surprised which bothers me further. Thank you for your time!