Is it permissible to suffice with slaughtering one large sheep for the aqīqah of the newborn, instead of two sheep, and what should be done after a year has passed since his birth?

You are not required to slaughter another sheep, and what you have already done is sufficient. The more complete act is to slaughter two sheep for a boy, based on the Prophet's (peace be upon him) saying: "For a boy, two equivalent sheep, and for a girl, one sheep." However, one sheep is sufficient for a newborn, whether male or female.