A generating series for Murakami-Ohtsuki-Yamada graph evaluations

Murakami-Ohtsuki-Yamada introduced an evaluation of certain oriented planar trivalent graphs with colored edges. This evaluation plays a key role in the evaluation of the colored HOMFLY polynomial of a link in 3-space and its Khovanov-Rozansky categorification. Our goal is is to give a generating series formula for the evaluation of MOY graphs, which may be useful in categorification, and in the study of q-holonomicity of the colored HOMFLY polynomial.