Is an expiation required of one who ran over his son with a car, and what is it?

If a father accidentally runs over his son with his car, leading to the son's death, the father is subject to two obligations:

1. Blood money (Diyya): This is borne by the killer's 'aqila (male relatives on the paternal side) and is paid to the son's heirs. The father does not inherit anything from it. 2. Expiation (Kaffara): This involves freeing a believing slave. If one cannot be found, the father must fast for two consecutive months, as Allah Almighty says: ﴿And it is not for a believer to kill a believer except by mistake. And whoever kills a believer by mistake – then freeing a believing slave and a compensation (Diyya) delivered to his family, unless they remit it as charity. But if he was from a people hostile to you and he was a believer – then freeing a believing slave. And if he was from a people with whom you have a treaty – then a compensation (Diyya) delivered to his family and freeing a believing slave. And he who finds not [the means to do so] – then a fast for two consecutive months as repentance to Allah. And Allah is ever Knowing and Wise.﴾ [An-Nisa: 92].

However, if the father running over his son does not result in death, but only injuries:

If the required compensation for the injuries amounts to one-third of the diyya, it is borne by the 'aqila and paid to the injured son. If the required compensation is less than one-third, according to Abu Hanifa and the correct view in the Shafi'i school, it is borne by the 'aqila. However, the relied-upon opinion is that it is borne by the perpetrator himself (the father), based on the statement of 'Umar, may Allah be pleased with him, who ruled regarding diyya that nothing from it is borne until it reaches the compensation for a "mamooma" (a head injury that exposes the brain) (i.e., one-third of the diyya). There is no expiation (kaffara) on the father in either of these cases where the injury does not lead to death.