Minecraft – Sliding blocks with pistons – can I delay the rising edge only

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

Here's a working concept of what you're probably looking for:

The trick is to split the output with varying delays, then each of them take their turn keeping the gate lifted up. The more rows of repeaters you have (with increasing length), the longer the button's signal.