Captain reported severe nose wheel vibration on take off. The flight crew elected to divert for a precautionary landing.

Narrative

Severe nose wheel shaking at V1/rotation speed on takeoff Runway XXR ZZZ. Above landing weight. Above 10;000 ft.; we consulted with Maintenance and Dispatch and diverted to ZZZ1 (ZZZ1 just far enough for below max landing weight)-stated reason not to continue to ZZZ2 was no tires available in ZZZ2. Upon landing I was very gentle with the nose gear and kept it off until as slow as possible- upon spinning up; it was less violent at a slower speed and some weight off; but very obviously had something wrong with it. All seemed normal at taxi speed-no vibration. I have over the years had occasions of tire out of balance and nose wheel shimmy from time to time. This was an order of magnitude worse-as in a factor of 10. The whole front of the airplane shook; flight deck instruments were difficult to read; and the front Flight Attendant and Passengers were very aware. This begs the question of a tire manufacturing flaw or of an improper tire change. We were delayed originally as Maintenance announced a tire change in ZZZ just as the first passengers showed up to board- passengers sent back to gate. So this was the first flight after a nose tire change. I request a follow up email describing the extent of the problem- bad tire; bad tire change; or something else. We landed about 100-200 lbs. below max landing weight in ZZZ1. As we were too heavy to land immediately and all was normal in flight; I took my time getting above 10;000 ft. and discussing with FO (First Officer) before contacting Maintenance. This was the FO first diversion and it was a very hectic time for him. ZZZ1 comes fast when not planned. I slowed the airplane a bit to give him more time and tried to make everything as easy as possible and as much like a normal flight as possible. Definitely good to use these types of things in CQ (Crew Qualification) training. Getting away from the expected makes everything harder and more challenging.Dispatch; Maintenance; Operations all very helpful. It is a pleasure working with all these excellent people during an abnormal situation.