Take my post with a big pinch of salt, I may be entirely wrong here (and welcome anyone to correct me entirely!)
First thing to note: in the paper they are using quadratic splines, not cubic ones. As a result the numbers of their control points and knots won't match the configuration you have above.
I 'think' this is their equivalent of their cox-de-boor (ish) for a triangle. I'm guessing the 3 terms on the right [B(u|K\v0), B(u|K\v1), and B(u|K\v2)] are simply the standard cox-de-boor algorithms?
B(u|K) = a0B(u|K\v0) + a1B(u|K\v1) + a2B(u|K\v2)
If a0, a1, and a2 are just the barycentric coordinates of the point being tessellated, then I am assuming the 3 basis functions would look something like this:
a1 = v