# Coefficients of the composition of an analytic function

Suppose we have a function $$f(x):\mathbb{R}\rightarrow \mathbb{R}$$ which is analytic. We can write its Taylor expansion around $$x=0$$ as: $$f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$$.

The function $$f^{(k)}(x) = f(x)\circ\dots\circ f(x)$$, that is the composition of $$f(x)$$ with itself $$k$$ times, is also analytic as a composition of analytic functions. So we can also write its Taylor expansion around $$x=0$$ as $$f^{(k)}(x)=b_0 + b_1x + b_2x^2 + b_3x^3 + \dots$$.

My question: is there a way to get a close form of the coefficients $$b_0,b_1,\dots$$ in terms of $$a_0,a_1,\dots$$?

In particular, suppose the coefficients of $$f(x)$$ goes to zero as $$O\left(\frac{1}{n!}\right)$$ (e.g. $$f(x)=\exp(x)$$), how fast the coefficients of $$f^{(k)}(x)$$ will go to zero?