Air carrier Captain reported an NMAC during cruise with a descending aircraft. The flight crew took immediate evasive action that resulted in mach buffet.

Narrative

We were at FL 370 cruising with the autopilot on; FO's (First Officer) leg. Beautiful weather; no issues; nice flight. approximately 60 miles west of ZZZZZ intersection my FO and I both simultaneously noticed a TCAS target on the MFD that was changing from white to yellow out in front of us on the magenta line. The altitude started to be depicted as 1;000 above and descending; 900; 800; 700. At that point the FO and I agreed this guy was coming at us and we got ready to disengage the autopilot. It got to 600 and that's when I believe the RA started. I remember seeing 500 above in red on the target and then transferring my attention to the PFD to get us out of the red bars. The FO had initiated the descent properly having kicked the autopilot and autothrottle off while I monitored. At the same time; the controller started yelling at the aircraft coming towards us; this all got really bad really fast. The other aircraft did not stop descending. The Controller yelled at him to stop the descent and turn right forty five degrees immediately. Our TCAS changed to increase descent; we had to lower the nose more; while reducing power as well. Aircraft did not overspeed but we did get mach buffet as the engines didn't spool down quick enough. We kept pushing down while the Controller continued to dialog with this aircraft; there was questioning coming from the aircraft about a descent clearance rather than stopping the descent. The Controller was in what I would call a state of mind based on his tones that he was expecting us to hit each other. I won't soon forget that feeling; this was like no other RA I've experienced in my 23 years of 121 flying. We got down to FL360 and by then the other aircraft started to climb. Eventually we did get the clear of conflict. No aircraft limitations were exceeded; no write ups made. Checked with the crew in the back they were OK. Made an announcement to the passengers.I have had RA's before; both in training and in real life. This RA absolutely rattled both myself and the FO. The offending aircraft did not stop descending; the Controller was lost as to what the situation was and didn't realize what was happening until well into the RA. Both my FO and I spoke at length about this afterwards; we thought we might not make it home that day in the middle of it and it scared the hell out of us both. The last thing I saw was 500 above but I do believe it was less due to his rate of sink and when I gave up looking at the MFD to stay on the PFD's escape guidance which was being increased to more nose down.We both called the Operations Manager on landing and removed ourselves from the rest of the trip. I have the N number of the aircraft if needed. I also want feedback and phone calls about this from someone explaining where the error occurred - ATC or the aircraft. Both my FO and myself need this information; this was a pretty unsettling event. On the plus side I will say the training we get and the TCAS system itself did exactly what it should have.