A321 Captain reported an NMAC during takeoff requiring evasive action.

Narrative

On rotation; ZZZ runway XXL; I noticed; as the PM; an aircraft that looked about 1000' AGL flying N-S down the beach in our lateral flight path. Based on the line of sight I could tell that our current climb rate would take us over the top so we continued normal operations. At about 1300' we received a TA on the aircraft and it was -300' below us and was disappearing under the nose. I kept sight of it until it went under the nose and I was comfortable we were well above and past the aircraft. After getting out of sterile cockpit I discussed with the FO and we were both 1. Surprised that the aircraft was there 2. Surprised that tower didn't give us a point out. Our normal departure climb kept us clear but if we had an engine problem and had to level off at 1000' the aircraft would have been a factor. During the event I was able to peek down at the CDTI and the aircraft was showing up which again makes me wonder why no one said anything to us. I checked the ZZZ tower airspace chart; the Jepps books; and company pages for any reference to 'look out for VFR aircraft flying down the beach' but couldn't find anything. The FO and I are both positive no one ever called out the traffic to us. Good reminder that even on a clear VFR day under IFR control looking out the window to clear is a must.Give point outs to aircraft that could conflict with departure traffic.