Greetings,,,
I'm sure you can find lots of answers on this. You could build any of the various microcontroller units that can run a simple off and on sequence, or even use a simple counter and have it change positions on outputs from 1 to 16 to go on and off.
Some of this can get really technical with all the different options. How many different combinations do you need, etc. If you only have to turn on and off the channels in order, 1 to 16, with only one on at a time, or different combinations.
You could even use a mechanical turning tin can, with extruded parts that turn switches on. Some of the other boards have designs that people have made to run things with outputs like what you want. I'll have to go back to check to point you to something already designed.
The main thing is to make sure any AC controlling aspects are properly built, in order to maintain safety.
Keith