Is expiation required for someone who swore an oath in jest and did not tell anyone about what he swore, while not remembering his intention at the time of the oath, and whether it was a 'laghw' (unintentional) oath or a 'mu'anqadah' (binding) oath?

If you are certain that you swore by Allah or by one of His attributes that you would inform someone about your brother and you did not do so, then you have broken your oath, and the expiation for breaking an oath is incumbent upon you. The expiation for breaking an oath is to feed ten poor people or clothe them, or to free a slave. If you cannot find any of these, then you must fast for three days. However, if you are in doubt about the oath or its validity, then nothing is incumbent upon you, because the default is the acquittal of responsibility. It appears that you suffer from obsessive thoughts, so you must disregard them and rely upon Allah.