Does anyone have a nice general formula for the sum of velocities?

We know that the sum of two velocities \(x,y\) is \(\dfrac{x+y}{1+\dfrac{xy}{c^2}}\) where \(c\) is the speed of lights. I went ahead and calculated the sum of three velocities \(x,y,z\), expecting to get something like \(\dfrac{x+y+z}{1+\dfrac{xyz}{c^3}}\) but instead I got \[\dfrac{x+y+z+\dfrac{xyz}{c^2}}{1+\dfrac{xy+yz+zx}{c^2}}\]. I went further to calculate the sum of \(4\) velocities \(w,x,y,z\), to get the ugly expression \[\dfrac{w+x+y+z+\dfrac{wxy+xyz+yzw+zwx}{c^2}}{1+\dfrac{wx+wy+wz+xy+xz+yz}{c^2}+\dfrac{wxyz}{c^4}}\]

So can anyone else generalize? I do not see an obvious pattern yet, except for the fact that those different terms look suspiciously like something in Vieta's Formulas...

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