Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$.
I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$.
Let $\lambda$ be a partition of $n$, i.e. a tuple of numbers $(\lambda_1, \ldots, \lambda_k)$ satisfying $\lambda_1 + \ldots + \lambda_k = n$. We introduce the shorthand $f^{(\lambda)}$ for the product $f^{(\lambda)}(x) = f^{(\lambda_1)}(x)\cdots f^{(\lambda_k)}(x)$
It is easy to see that the derivatives of $g$ can be expressed in the form
$$g^{(n)}(x) = (\sum_{\lambda} c_\lambda f^{(\lambda)}(x))g(x)$$ where $\lambda$ runs over all partitions of $n$, e. g.:
$$g^{(2)} = (f^{(2)} + f^{(1)}f^{(1)})g$$ and $$g^{(4)} = (f^{(4)} + 4f^{(3)}f^{(1)} + 3f^{(2)}f^{(2)} + 6f^{(2)}f^{(1)}f^{(1)} + f^{(1)}f^{(1)}f^{(1)}f^{(1)} )g$$
So $c_{(2)} = c_{(1, 1)} = 1$ and $c_{(4)} = c_{(1, 1, 1, 1)} = 1$, $c_{(3, 1)} = 4$, $c_{(2, 2)} = 3$ and $c_{(2, 1, 1)} = 6$.
The question is: what are the numbers $c_\lambda$? Do they have name? Can they be easily computed from the Young tableaux? I hoped that by computing a few terms I could spot the pattern, but no luck...