Below is the simulation of a power-on sequence circuit that uses a CD4015 4-bit shift-register and a CD4049 inverter.
The power switch is simulated by generator V1 controlling switch S1, pulsing at 1 pps.
As you can see it continuously sequences through the power on sequence and then off in the reverse order. (The green and red traces are vertically offset to make them more visible. The are all the same amplitude.)
D1, R2, and C2 form a switch debounce circuit.
C1 and R1 are a power-on circuit to reset all outputs to the zero state at power on.
Vcc can be any DC power from 3V to 15V.
The Pwr_On output from pin 5 can be used to drive a mechanical or solid-state relay to control the 120Vac. This will require a buffer transistor to drive the relay. If you need advice on that, I can provide some.
Note that this circuit requires power independent of the 120Vac amp power so you may need a small wall-wort supply for that.
Not that
all unused inputs on both packages must be connected to ground (not shown) or the circuit may become erratic.