Child Support Question

What rights does the father have if his name is not on the birth certificate? Does the father still need to pay child support? The reason I ask is the man got another woman pregnant. He doesn't want the wife to find out about his unborn child and has told the woman he can't be in the child's life.

***The father lives in Tennessee and does not want to be in the child's life.

Should the single mother hold him accountable since it was both of them that made the child?

The mother doesn't qualify for assistance from the state. She is a couple of hundred above the amount she would need to be under and has no family support.