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According to Roman Law, the child of a male slave or of a free man and a slave woman was a slave like his/her mother - the status of slave passed from mother to child. Thus the child of a freedman would be either born free or born a slave depending on the status of his/her mother.
Freedmen had the status of Roman citizens - though second class citizens still partially dependent on their former masters. A male child born to a freedman and a free mother - even if she may once have been a slave - was a free born Roman citizen, though usually of low status.
As I remember, Emperor Publius Helvius Pertinax Augustus, was born to a freedman Helvius Successus and rose in society to be a military officer, a senator, a governor, a consul, and finally emperor.
Emperor Marcus Opellius Severus Macrinus Augustus is also said to have been the son of a freedman according to some sources.