Thermistors - Important !
Using thermistors rather than resistors is a common question, and while there are many caveats they will generally work well. Unfortunately, it can be very difficult for the novice (and not-so-novice) to determine the proper value and size, and manufacturers don't help much. The specification format from one maker rarely matches that of another, and making direct comparisons is rarely easy. Some quote a maximum current, others a rating in Joules, and some include almost nothing except the nominal resistance at 25°C and the dimensions - hardly helpful.
Many people like the idea of using NTC (negative temperature coefficient) thermistors for inrush limiting, with a common claim being that no additional circuitry is needed. In a word, DON'T. This is possibly controversial, because they are used by a number of major manufacturers so must be alright - or so it might seem. If used in a switched system as described here, they are safe enough, but I have personally seen (yes, with my very own eyes) NTC thermistors explode mightily if there is a fault. Resistors can also fail, but the failure is (usually) contained - there are exceptions of course. In general, NTC thermistors are designed for very high peak current, but as noted earlier, you will see many different ways to describe the same thing, with almost no commonality between makers.
If the relay fails to operate because you didn't listen to me and used the amp's supply, the thermistor will (in theory) become a low resistance due to the current flow and the fuse will blow. However, if current is too high due to a major fault, the thermistor may explode before the fuse has a chance. I'm unsure why some people insist that the thermistor is somehow 'better' than resistors - it isn't, and in some cases may even be a less robust solution. As noted below, a resistor (or thermistor) value of about 50 ohms (230V) or 25 ohms (120V) is a pretty good overall compromise, and works perfectly with transformers up to about 500VA. The resistance should be reduced for higher power transformers.
If a thermistor is used, it needs to be sized appropriately. While some small thermistors may appear quite satisfactory, they will often be incapable of handling the maximum peak current. I suggest that you read the article on inrush protection circuits for more information. A suitably rated thermistor can be used in any version of this project (including the PCB based unit shown in Figure 6).
Under no circumstances will I ever suggest a thermistor without a bypass relay for power amplifiers, because their standby or low power current is generally insufficient to get the thermistor hot enough to reduce the resistance to a sensible value. You may therefore get power supply voltage modulation, with the thermistor constantly thermally cycling.
If there is enough continuous current (Class-A amplifier for example), the surface temperature of any fully functioning thermistor is typically well over 100°C, so I consider bypassing mandatory to prevent excess unwanted heat. A bypass circuit also means that the thermistor is ready to protect against inrush current immediately after power is turned off. Without the bypass, you may have to wait 90 seconds or more before it has cooled.