Iterated cyclic exponentials and power functions with extra-periodic first coefficients

If $f$ is the iterated $m$-cyclic exponential
$$ f(z)=e^{\lambda\alpha_1ze^{\alpha_2ze^{\dots}}}= \langle e^z;\lambda\alpha_1,\alpha_2,\dots,\alpha_m,\alpha_1,\dots\rangle, $$
where the first coefficient, $\lambda\alpha_1$, in the sequence of coefficients is extra-periodic, then in its power series expansion at $z=0$, $\sum_{n=0}^\infty\frac1{n!}H^{(n)}(f) z^n$, the form $H^{(n)}(f)$ can be written as
\begin{align*} H^{(n)}(f) &=\lambda\alpha_1\sum_{k_1+\dots+k_m=n}\frac{n!}{k_1!\dotsb k_m!} (k_1\alpha_2)^{k_2}(k_2\alpha_3)^{k_3} \\ &\qquad\times\dots\times(k_{m-1}\alpha_m)^{k_m}[(k_m+\lambda)\alpha_1]^{k_1-1}. \end{align*}
This formula is generalized to any number of extra-periodic coefficients at the start of the sequence. It is also shown that in some cases iterated cyclic exponentials whose first coefficients are not elements of the $m$-cyclic sequence $(\alpha_1,\alpha_2,\dots,\alpha_m,\alpha_1,\dots)$ can furnish a solution of a first-order system of differential equations with rational right-hand side.
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