What is the expiation for one who swore, "I have slept with my mother if I do not do such-and-such," and is this oath permissible?

Swearing by other than Allah, such as saying, "No, by the Kaaba," is forbidden, due to the Prophet, peace and blessings be upon him, saying: "Whoever takes an oath should swear by Allah or remain silent," and his saying: "Whoever swears by other than Allah has disbelieved or associated partners with Him." This oath is not binding and does not necessitate an expiation, but it is a sin that requires seeking forgiveness and repentance.