The mother claims them do I put down that I support them?
No.
If you are not the custodial parent and have not received a signed Form 8332 from the custodial parent releasing their claim to the child's personal exemption, then you do not claim the children as dependents on your tax return. Child support payments are not reported on a tax return.
There is a way to split the tax benefits. This may be helpful in your negotiations with the ex:
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.
Ref: <a rel="nofollow" target="_blank" href="http://www.irs.gov/publications/p17/ch03.html#en_US_2014_publink1000170897">http://www.irs.gov/publications/p17/ch03.html#en_US_2014_publink1000170897</a> Scroll down to "Children of divorced or separated parents (or parents who live apart)"