What is the expiation for someone who, out of negligence, left the chicken coop open, leading to cats preying on some of the chickens?

You must repent and seek Allah's forgiveness for the negligence that led to the death of the chickens due to cats. There is no specific expiation for this. Repentance is prescribed for a Muslim in all circumstances, even for what is thought or doubted to be a sin, based on Allah's saying: "And turn to Allah in repentance, all of you, O believers, that you might succeed." General repentance forgives all sins unless there is an impediment that specifically limits it.