Air carrier Captain reported the planned DG Summary arrived during gate departure without the Final DG report. During flight the First Officer queried Dispatch which resulted in a Final DG report.

Narrative

On this flight we released the parking brake and pushed back without the Final Dangerous Goods Summary; and without having sent the dangerous goods acknowledgement code message on ACARS. This was the final leg on a 2 day trip. We arrived at the aircraft 1 hour late due to weather on the previous leg; and the flight was already loaded with passengers and flight attendants. Initializing the ACARS produced a Planned Dangerous Goods Summary. Pilot preflight duties were performed normally. Aircraft was closed up and jet bridge pulled. When doors were being armed; the door at 4L wouldn't arm; so the jetbridge was pulled back to the aircraft and Maintenance repaired the issue. We awaited the new Maintenance Release. The Final DG summary never showed up; but we missed that fact due to all the irregularities to this point. We pushed back and departed. The First Officer (FO) queried me on climbout if we had received the final summary; which we had not. We contacted both station operations and load planning to determine if DG had actually been loaded; or only planned. They responded with a printout of the Final Dangerous Goods via ACARS.