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Was a court order given for you to pay child support or medical support of the child?
Was any insurance used when the child was admitted?
If there is no order for support and you do not have custody of the child then the father would be held liable as well as you. Parents are held jointly responsible for any health bills when a order has not been established.
If you pay this you can then take your ex to court and ask that the other half be paid by him to you as reimbursement of the bill.
I understand the situation, but the law requires that both parents are responsible for the bill if no order is in place. I know this is frustrating but in the end children do get sick and when they do both parents are responsible for the bill regardless of the situation between the two if there is no order.
I am sorry, I know this isn't the answer you were hoping for. I do not make the laws nor do I agree with them all.