The omission may have been accidental or it may have been deliberate. Regardless of the circumstances, if you are the father of a child, you have the legal right to have your name listed as the "Father of Record".
In the United States, Birth Certificates are classified as public records. This means that you may request a copy without having to specify a reason, nor do you need to have any legal documentation giving you the "right" to a copy of the Birth Certificate.
REQUEST FOR BIRTH CERTIFICATE RECORDS
(name of person signing)
hereby request a copy of the Birth Certificate for [child's name], born on or about [child's birth date], to [mother's name] whose last known address is [mother's address, if known].
If you are unable to comply with this request, please send me a letter in writing, quoting the State's civil or legal code that supports your decision. To my knowledge, birth certificates are public records and therefore I am within my rights to request a copy.