If you're not on the birth certificate - and you're not, since you two didn't fill out a Voluntary Acknowledgment of Paternity - you have NO rights to the child AT ALL until paternity is legally established.

I'm afraid that means you're going to have to avail yourself of the courts. Your first step is to begin an action to establish paternity. You can commence this action by filing an Application for Establishing Paternity through the Oregon Child Support Program.

After paternity is established, you can pursue a visitation order and arrange for child support.