In this question I asked for a closed-form solution to this functional differential equation: \begin{align*} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align*}
It doesn't look like there is one, but I'm interested in another question about $f$. The Maclaurin coefficients $f_n = [z^n]f(z)$ satisfy
\begin{align*} f_n &= \frac{e^{-n-1}}{n} \sum_{j=1}^n f_{j-1} f_{n-j}, \qquad n > 0, \\ f_0 &= 1. \end{align*}
This is a full-history recurrence relation for $f_n$; that is, all previous terms are involved. Is there a finite-history recurrence?
A sufficient condition is that $f(z)$ is holonomic, but that's not clear to me.