Indeed, very Interesting! @J. M., this is a neat way to define @Bill's products. I typeset them via a matrix here for clarity:
kProduct[k_] :=
Divide@@Array[{{512 #^3,0},{32 #^3(42 # -37),(2 # -1)^3}} &,k,1,Dot][[All, 1]]
We can see k-th difference is log-linear in k:
ListLogPlot[Table[kProduct[k]-Pi,{k,10}],
PlotTheme->"Detailed",PlotRange->All,FrameLabel->{k,Log[Pi-kProduct]}]
