A problem of interest that has come up for me recently is solving the following.
There is a formula for a general $n$-th order derivative of a composition as shown above: http://mathworld.wolfram.com/FaadiBrunosFormula.html
In terms of the Bell Polynomials, we can write
And the Bell polynomials of the second kind are shown in the Wolfram link above. I am wondering if there is a closed-form solution for the sum of the series of Bell Polynomials.