All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
If you wrote down the divorce of your wife or signed the divorce paper, determining to divorce her, then the divorce took place. That is because the writing of divorce counts as a divorce when it is intended as we have pointed out in Fatwa 90056. But if you did not intend to divorce her, then divorce does not count.
If a man divorces his wife three times in one single utterance, the majority of scholars are of the view that the woman has become thrice divorced and it becomes forbidden for her husband until she has been married to another man in a serious marriage and then they separate by divorce. For more benefit, please refer to Fatwa 82541 about the kinds of divorce.
Other scholars are of the view that this is to be regarded as a single divorce. This was adopted by Shaykh Al-Islam Ibn Taymiyyah . However, the opinion which we adopt here in Islamweb is the opinion of the majority of scholars. For details, kindly refer to Fataawa 82330, 155573, and 94110.
Therefore, according to the opinion of the majority of scholars, your wife has become forbidden for you. If that child has not reached the age of discernment, then the mother has right to his custody, so long as she does not re-marry. If she re-marries, custody will be transferred to those who are more entitled to take care of him such as the grandmother (his mother's mother). For more benefit, please refer to Fatwa 84618 about the order of those who are more entitled to custody of the child.
Allaah Knows best.