I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for context) by hypothesizing and using the first few terms of the Baker formula to verify. In order to prove my result rigorously, I think I need the nth term of the Baker formula. Is there an expression for that - I could not find it through online research. ? It would also help if I could find a proof of the BCH formula which is based on recursion i.e. if I could see how the nth term relates recursively to the (n-1)th term. 



nth term in the Baker-Campbell-Hausdorff formula
The Dynkin formula is somewhat cumbersome. Maybe a better choice is Goldberg's version <http://projecteuclid.org/euclid.dmj/1077466673> In the commutator form Goldberg's result is reformulated in <http://www.ams.org/journals/proc/1982-086-01/S0002-9939-1982-0663855-0/> (Cyclic relations and the Goldberg coefficients in the Campbell-Baker-Hausdorff formula, by Robert C. Thompson).


By the way an interesting early history of the Baker-Campbell-Hausdorff-Dynkin formula can be found in <http://link.springer.com/article/10.1007%2Fs00407-012-0095-8> (The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin, by R. Achilles and A. Bonfiglioli). 


