Let's consider a function (or a way to obtain a formal power series):

$$f(x)=x+(x+(x+(x+(x+(x+\dots)^6)^5)^4)^3)^2$$

Where $\dots$ is replaced by an infinite sequence of nested brackets raised to $n$th power.

The function is defined as the limit of:

$$f_1(x)=x$$

$$f_2(x)=x+x^2$$

$$f_3(x)=x+(x+x^3)^2$$

etc.

For $|x|$ 'small enough' we have a finite limit $f(x)$, but I'm not really interested in it right now.

What I'm interested in - if we consider the function to be represented by a (formal) power series, then we can expand the terms $f_n$ and study the sequence of coefficients.

It appears to converge as well (i.e. the coefficients for first $N$ powers of $x$ stop changing after a while).

For example, we have the correct first $50$ coefficients for $f_{10}$:

$$(a_n)=$$

`0,1,1,0,2,0,1,6,0,6,6,24,15,26,48,56,240,60,303,504,780,1002,1776,3246,3601,7826,7500,18980,26874,38130,56196,99360,153636,210084,390348,486420,900428,1310118,2064612,3073008,4825138,7558008,11428162,18596886,26006031,43625940,65162736,100027728,152897710,242895050,365185374`

I say they are correct, because they are the same up until $f_{15}$ at least (checked with Mathematica).

Is there any other way to define this integer sequence?

What can we say about the rate of growth of this sequence, the existence of small $a_n$ for large $n$, etc.?

(see numerical estimations below)Does it become monotone after $a_{18}=60$? (Actually, $a_{27}=7500$ is smaller than the previous term as well)

(see numerical estimations below)Are $a_0,a_3,a_5,a_8$ the only zero members of the sequence?

(appears to be yes, see numerical estimations below)

The sequence is not in OEIS (which is not surprising to me).

**Edit**

Following Winther's lead I computed the ratios of successive terms for $f_{70}$ until $n=35 \cdot 69=2415$:

$$c_n=\frac{a_{n+1}}{a_n}$$

And also the differences between the successive ratios:

$$d_n=c_n-c_{n+1}$$

We have:

$$c_{2413}=1.428347168$$

$$c_{2414}=1.428338038$$

I conjecture that $c_{\infty}=\sqrt{2}$, but I'm not sure.

- After much effort, I computed

$$c_{4949}=1.4132183695$$

Which seems to disprove my conjecture. The nearby values seem to agree with this.

$$c_{4948}=1.4132224343 \\ c_{4947}=1.4132265001$$

But the most striking thing - just how much the sequence stabilizes after the first $200-300$ terms.

How can we explain this behaviour? Why does the sequence start with more or less 'random' terms, but becomes monotone for large $n$?

**UPDATE**

The sequence is now in OEIS, number A276436