Should I let him be added now??

Ok so my son was born this month and he’s Dad wasn’t there didn’t show any interest and still hasn’t since he’s new relationship with his girlfriend. He’s been partying and drinking lives with her and her kids, her baby daddy also lives there too..It’s a messed up situation honestly. He hasn’t been around me all though the pregnancy only been to one appointment and never texted me for months to ask how I was..At first he questioned the pregnancy and the. Kinda accepted it..Anyway my son is 4 weeks now I had an emergency c section back in January he knew my due date was the 17 but my son came on the 11. He was never called (I wasn’t aware of this because I was still out after surgery) My family never seen a point to call him and apparently his # wasn’t working. He’s sent one message on Facebook..hasn’t asked about baby hasn’t bought anything at all. I’m so tired of my son being done this way. He’s not on the birth certificate because I feel he don’t deserve to be since he can’t be there in his life..He takes pictures with his girlfriend boys but don’t seem to worry about his own son..Am I wrong for feeling this way ?? Should I give it and contact him again? Even though he doesn’t seem worried..IDK🤷‍♀️