Half-iterate exp(z)-1: hypothese on growth of coefficients

[Leo.W]

expanding \(e^t-1,f'(t),f(t)^{n+1}\) and allocate the coefficients of \([t^{-1}]\) we can arrange and get a reccurence.
Then the recurrence should prove the asymp of a_n.

What is \([t^{-1}]\)?

It's blanket notation, only for convenience to represent the coefficient of a specific term, for example let \(f(z)=z^3+2z^2-z+5-\frac{\pi}{z^2}\)
then \([z^3]f(z)=1\)
\([z^2]f(z)=2\)
\([z^1]f(z)=-1\)
\([1]f(z)=5\)
\([z^{-2}]f(z)=-\pi\)