The Bae 146 requires the APU to be left on after start as it is the source of bleed air for A/C & pressurisation until approx. 1,500 feet, at which point the engine bleeds are turned on and the APU shutdown, it is also started on descent and carries the bleed air load on final approach.
Most other jets have sufficient bleed air available so the APU is generally shut down after the last engine is started.