If a father's name does not appear on the birth certificate, he does not have any legal rights to the child, according to Law Info of Texas. He is not the legal father since his name has been omitted from the legal document. Because the father is not listed on the birth certificate he has no rights to custody, visitation, or paying child support. Go to http://www.fathers4kids.com/html/FathersRights.htm?article_id=41 for details.
It's called abandonment. If the mother makes reasonable attempts to find him within a certain period then yes his rights are forfeited. But not having his name on the certificate does not automatically eliminate his rights. However, a Letter of Abandonment would clarify it if the mother is trying for sole custody and eliminate his parental rights (and her rights to child support.) A judge would clarify any gray areas.