To derive it from the wave function, see

https://farside.ph.utexas.edu/teaching/315/Waveshtml/node25.html#s5.3
Starting with EQN 292 is where you see the strength of each mode when the string is plucked at the center.

= 2A * 2^2 * sin(m pi/2) / (m ^2 * pi ^2)

= A * 8 * sin ( m * pi/2) / (m^2 * pi^2) (eqn A)

Our Capytalk is

((!Position * m) normSin / m squared * 8 / Float pi squared)

where normSin = sin (!Position * m * pi)

so

8 * sin (!Position * m * pi) / (m^2 * pi^2)

If Position = 0.5 (plucked at the center), then

8 * sin (m * pi * 0.5) / (m^2 * pi^2) same as (eqn A)

That's where the 8 came from and the divide by pi^2.

The (8 / pi^2) constant makes it so when you add up the mode amplitudes, they sum to 1. You could probably just ignore it when reasoning about the amplitudes. That leaves you with

sin (m * pi * 0.5) / m^2

so the mode strengths fall off with the square of the mode number (sort of like a triangle waveform).

Imagine when you pluck the string, you are pulling it out of alignment, creating a triangle shape. Depending on where you pluck it, the triangle is either asymmetric or symmetric (close to either fixed point or close to the center). If you do a fourier analysis of that triangular displacement shape, you get the strengths of the modes when you pluck the string at that position.