Were you married to mom? If so, the child would have automatically been considered yours anyway. If not, the burden is on you to ensure paternity before signing the birth certificate or acknowledgement of paternity.
Is the actual father ready, willing, and able to step up to establish himself as the father? The courts want children to have TWO parents, and if you've already claimed to be one of them, the court may not remove you even if you are not the biological parent without the "actual" parent to put in your place.
How old is the child now?
This isn't likely to involve mom's checkbook, but your own. As JK noted, if you had reason to believe that you were the father, then so did mom - so mom's not going to bear the brunt of costs for a wrong assumption. Unless you were married to her, you really had no rightful expectation of fidelity, so unfortunately the burden in such cases rests with you to make sure that you are the father BEFORE admitting to such on a legal document such as a birth certificate, AOP, or similar.

