What are my rights as a unmarried father in Washington state? My name is on the birth certificate.

Can you be more specific as to what you want to know?

You share custodial rights with the mother. You need to go to court for a custody & visitation order (which can be agreed to if the mother and you can agree). If she has actual physical custody, you can be required to pay child support (and vice versa).

You have the same rights as any father. Being on the birth certificate means you are the legal father. You also have the same responsibilities.