Minecraft – Enabling one of 16 outputs for activation with a binary 4-lever input

If you watch at about 2:50, you can see most of the left half of the circuit. The part for the stairs looks like it isn't shown very well but that is less important timing wise. Watch carefully over and over and see if you can replicate it this way. It looks like the repeaters for the middle activate and deactivate immediately. So the order (from a circuit standpoint) for both directions is M, L+R. This works for closing because the middle repeaters don't have anything to activate until the blocks are pushed into the middle. When opening, they will retract the middle immediately.

Breaking it down:

Opening

1. Middle repeaters deactivate (middle pistons retract)
2. Delay
3. Left and Right redstones deactivate (left and right pistons retract)

Closing

1. Middle repeaters activate (nothing happens because the middle pistons are not in position)
2. Delay
3. Left and Right redstones activate (left and right pistons extend)
4. Now the middle pistons have been pushed into position and are activated by the middle repeaters

Lightweight version
It seems I overcomplicated things a bit on the first attempt, this version should be better and more compact, at least for smaller numbers (first example produces 4 pulses). It's also possible to connect the "start" for another pulse straight from the second torch).

For this setup to work with a high pulse count you will need to put repeaters every 15 blocks in the redstone line, and for each of those repeaters there will need to be extra ticks between all stone blocks. This exact setup will work for your 11 pulses (last 6 blocks of 11 shown):

Heavy version (no limit)
Basically an outer "zig-zag" circuit that will send a new pulse straight through the middle line each time it passes a stone block. I tested it with up to 11 pulses, and you can just extend it as much as needed. Obviously it gets quite clumsy with a large number of pulses.