All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
A father must be prudent as regards taking care of his children since, in Islam, he is their guardian and they are his subjects; therefore, he isn`t permitted to neglect that which preserves their life and health. Since you-asker- haven`t hospitalized your son out of ignorance and by mistake, we pray that Allah Accepts your repentance, and Pardons you in case you sought His Forgiveness and confessed your negligence. Allah, The Almighty, Said {what means}: " He is the One that accepts repentance from His Servants and Forgives sins: and He knows all that ye do" {Al-Shura}.
Based on the analogical reasoning of the Muslim scholars, you don`t have to offer expiation and Diya (blood money) for two reasons:
First: You haven`t performed a direct action, nor performed one that caused his death.
Second: Recovery isn`t guaranteed, and his passing away, based on the physician`s diagnosis, isn`t definitive. Al-Khateeb Ash-Shirbini (May Allah be pleased with him) said: "If a starving person wasn`t able to take food from a certain person, and passed away as a result, then the one who denied him the food isn`t held liable before Allah because he didn`t commit an act that led to his death; however, he is considered sinful." {Moghni Al-Mohtajj, 5/373}.
A Hanafi book stated: "If a person was inflicted with a disease, and the physician told him that blood must be taken out of his system, but he didn`t undergo any medical procedure to that end, and died, then he (the sick) isn`t liable before Allah because he wasn`t certain of recovery. In comparison, this person differs from the one, above, who was hungry but didn`t eat although he could have, until he passed away. The difference is that had he eaten as much as he needed, he would have recovered, but since he hadn`t, then he killed himself, unlike medical treatment." {Al-Moheet Al-Borhani pp.373 vol. 5}. And Allah knows best.