A Formula for the Jones-Wenzl Projections

I present a method of calculating the coefficients appearing in the Jones-Wenzl projections in the Temperley-Lieb algebras. It essentially repeats the approach of Frenkel and Khovanov published in 1997. I wrote this note mid-2002, not knowing about their work, but then set it aside upon discovering their article. Recently I decided to dust it off and place it on the arXiv --- hoping the self-contained and detailed proof I give here may be useful. It's also been cited a number of times, so I thought it best to give it a permanent home. The proof is based upon a simplification of the Wenzl recurrence relation. I give an example calculation, and compare this method to the formula announced by Ocneanu and partially proved by Reznikoff. I also describe certain moves on diagrams which modify their coefficients in a simple way.