How is the deceased father's bank account divided between the mother and the children, and is the father's intention to include his children before his death taken into consideration during the division?

If your father died before including you and your sister as partners in his wealth, then you are only entitled to your legal share of the inheritance. Regarding your mother's inclusion in his wealth during his lifetime, this is to be considered: if it was a valid gift, fulfilling its conditions (that he gifted her the money when he was not suffering from a life-threatening illness, that she took possession of it when he was not suffering from a life-threatening illness, and that the gifted amount was known), then your mother is entitled to what her husband gifted her, and the remainder is divided among all of you as inheritance. However, if the gift did not fulfill its conditions, then your mother is not entitled to any of it, and all the wealth is divided among all the heirs according to Islamic law. A gift is not valid if it was made during a life-threatening illness, or if the mother was unable to take possession of the money until your father died, or if the gifted amount was undefined and its quantity unknown, according to the majority of jurists. As for the Malikis, they permitted gifting an unknown or undefined amount.