No.
If you (the custodial parent) claimed the child as a dependent, there is nothing the non-custodial parent can claim**.
The opposite is not true. If you allowed the non-custodial to claim the dependent, there are some things you (the custodial parent) could still claim.
There is a
special rule in the case of divorced & separated (including never married) parents.
When the non-custodial parent is claiming the child as a dependent/exemption/child
tax credit; the custodial parent is still allowed to claim the same child for Earned
Income Credit, Head of Household filing status, and day care credit. This
"splitting of the child" is not available to parents who lived
together at any time during the last 6 months of the year; then only one of you
can claim the child for any tax reasons. The tax benefits may not be split in any
other manner.
Note in particular
that the non-custodial parent can never claim the Earned Income Credit, Head of
Household filing status or the day care credit, based on that child, even when the custodial parent has released
the exemption to him.
So, it's good idea to let the other parent know that you
will be claiming those items, as many first time divorced parents are not aware
of this rule and may try to claim those items, which will cause the IRS to send
out letters.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
Scroll down to "Children of divorced or
separated parents (or parents who live apart)"
** The non custodial parent could claim any medical expense he paid for the child, but only as an itemized deduction (highly unlikely) but would not need the child's SS#.
