Faà di Bruno's formula. It will probably help you find a shorter formula.
For instance, where is the set of partitions of , like for . You can also group together in the sum the terms where the elements of the partition have the same size. This gives you what the coefficients are: in the expansion of , the coefficient before is the number of partitions of with blocks of size 1, blocks of size 2, etc.. (so that ). In other words, the coefficient is .
This is connected to Bell polynomials. The Wikipedia gives other formulations.