Recent references on the matter at hand include, a lecture slide [The Konvalinka-Amdeberhan conjecture
and plethystic inverses](https://people.brandeis.edu/%7Egessel/homepage/slides/K-A%20conjecture.pdf) and a preprint on [Counting tanglegrams with species](https://arxiv.org/pdf/1509.03867.pdf) by I. Gessel; the initial work [On the enumeration of tanglegrams and tangled chains](https://arxiv.org/pdf/1507.04976.pdf) by S. Billey et al.


For each prime $p$, let's consider the family of (integral) sequences
$$a\_p(n):=\sum\_{\lambda\vdash n}\frac{1}{z\_\lambda}\prod\_{i=2}^{\ell(\lambda)}(p\lambda\_i+\cdots+p\lambda\_{\ell(\lambda)}+1);$$
where the sum runs through all $p$-ary partitions $\lambda$, of $n$, and $\ell(\lambda)$ is the length of the partition and the numbers $z\_{\lambda}$ are well-known since the number of permutations in $\mathfrak{S}\_n$ with cycle type $\lambda$ is computed by $\frac{n!}{z\_{\lambda}}$.


On the other hand, Michael Somos proposed the functional equations $$
x\cdot A\_p(x)^p=\frac{x\cdot A\_p(x^p)}{1-p\cdot x\cdot A\_p(x^p)}
$$ and a couple of these are listed on OEIS as [A085748](https://oeis.org/A085748) and [A091190](https://oeis.org/A091190). However, there are not enough interpretations attached to $A\_p(x)$ or its coefficients on OEIS.


After some experimentation, I wish to ask:



> 
> **QUESTION.** Can the following be justified or refuted?
> $$A\_p(x)=\sum\_{n\geq0}a\_p(n)x^n.$$
> 
> 
> 


**Remark.** Observe that neither $A\_p(x)$ nor $a\_p(n)$ have been found to output integers, unless $p$ is a prime.



Tanglegrams and functional equations of M. Somos
In my lecture slides that Tewodros cites, I studied symmetric functions that I called $g\_m$, where $m$ is a positive integer, that have constant term 1 and satisfy  

$$ -L\_m[g\_m] = p\_1,$$
where $L\_m$ is the Lyndon symmetric function given by
$$L\_m = \frac{1}{m} \sum\_{d\mid m} \mu(d) p\_d^{m/d}.$$
Here $\mu$ is the Möbius function, the $p\_i$ are power sum symmetric functions, and $L\_m[g\_m]$ is the plethysm of $L\_m$ and $g\_m$. I showed that $g\_m$ is an integral symmetric function; i.e., its coefficients in the underlying variables are integers, and that if $m$ is a power of the prime $q$ then for any $\alpha$,
\begin{multline\*}
\quad
g\_m^{-\alpha} = 1+
\sum\_{n=1}^\infty \sum\_{\lambda}\frac{p\_{\lambda}}{z\_\lambda}
\times
\alpha\prod\_{j=2}^{l(\lambda)} (m\lambda\_j+m\lambda\_{j+1}+\cdots +m\lambda\_{l(\lambda)}+\alpha),\quad\tag{1}
\end{multline\*}
where the sum on $\lambda$ is over all partitions of $n$ in which every part is a power of $q$. (I'm using $q$ here instead of $p$ to avoid confusion with the power sum symmetric functions.) This shows that if $\alpha$ is positive and $m$ is a power of a prime then all coefficients of $g\_m^{-\alpha}$ are positive. I don't know that this is true if $m$ is not a power of a prime, though I suspect that it is.


Now let us specialize the symmetric functions (in the variables $x\_1, x\_2,\dots$) by setting $x\_1=x$ and $x\_i=0$ for $i>0$, or equivalently, $p\_i =x^i$ for all $i$. Let $G\_m(x)$ be the image of $g\_m$ under this specialization. Then $G\_m(x)$ satisfies the functional equation
$$-\frac{1}{m}\sum\_{d\mid m}\mu(d) G\_m(x^d)^{m/d} = x.$$
If $m$ is a power of the prime $p$, this may be written
$$G\_m(x^p)^{m/p} -G\_m(x)^m = mx$$
and it's easy to check that if $m=p$ then $A\_p(x) =1/G\_p(x)$, where $A\_p(x)$ is as in the original question. Moreover, it follows from $(1)$ that if $m$ is a power of the prime $p$ then for any $\alpha$ we have
\begin{multline\*}
\quad
G\_m(x)^{-\alpha} = 1+
\sum\_{n=1}^\infty x^n
\sum\_{\lambda}\frac{\alpha}{z\_\lambda}\, 
\times\prod\_{j=2}^{l(\lambda)} (m\lambda\_j+m\lambda\_{j+1}+\cdots +m\lambda\_{l(\lambda)}+\alpha),\quad\tag{2}
\end{multline\*}
where the sum on $\lambda$ is over all $p$-ary partitions of $n$. The case $m=p$, $\alpha=1$ answers the OP's questions in the affirmative.


In all cases $G\_m(x)$ has integer coefficients, but I can only prove that $G\_m(x)^{-1}$ has positive coefficients when $m$ is a prime power, when it follows from $(2)$, though this is likely true in all cases. The stronger statement that $1-G\_m(x)$ has positive coefficients also follows from $(2)$ when $m$ is prime power.


I am working on a paper with detailed proofs of these formulas, but it will take a while.


