Illegitimate minor children shall follow and have the settlement of their mother, unless the father is legally found or admitted by him to be such at the time and place of their birth, in which case they shall follow and have the father's settlement. If neither parent has a settlement, then it shall be in the county in which the minor child was born, if the birth occurred in this state.
Such minor children born in charitable or correctional institutions, or while the mother is legally an inmate thereof, whether on parole or leave of absence, shall be chargeable to the place of the settlement of the mother or father or from which they were admitted or committed, rather than to the place where that institution is located.