No. If you lived together unmarried with the other parent of your child, only one of you may claim the child as a dependent. That person should list the child as a dependent, and is eligible for all tax benefits, including the child, tax credit, and dependent care credit. The other parent should not list the child on the return at all to prevent problems.
potentially, your partner could list all the costs on their tax return, as if you gave them money, and they paid the expenses directly. This would be safer if that is what you actually did (and you had canceled checks are bank transfers to prove it) but I don’t think it’s likely to raise an issue if you are audited.
