closed form for these power series coefficients? The context for this is series approximation for perturbed iterations of the Mandelbrot set with arbitrary power $P$ (aka multibrot).
Define an arbitrary precision reference orbit:
$$Z_{n+1} = Z_n^P + C$$
Using lower case for low precision differences:
$$Z_{n+1} + z_{n+1} = (Z_n+z_n)^P + (C + c)$$
so:
$$z_{n+1} = \sum_{p = 1}^P \begin{pmatrix} P \\ p \end{pmatrix} Z_n^{P-p} z_n^p + c$$
Now approximate the difference in iterate $z_n$ by a power series in the difference in parameter $c$:
$$z_n = \sum_{k=1}^\infty A_{P,k,n} c^k$$
Combining the previous two equations gives:
$$\sum_{k=1}^\infty A_{P,k,n+1} c^k = \sum_{p=1}^P \begin{pmatrix} P \\ p \end{pmatrix} Z_n^{P-p} \left(\sum_{j=1}^\infty A_{P,j,n} c^j\right)^p + c$$
Equating coefficients of $c^m$ gives a collection of iteration formulas for $A_{P,m,n+1}$ in terms of the previous coefficients and $Z_n$. These iterations are defined implicitly by the previous equation.  The question is:
Is there an explicit closed form expression, perhaps in terms of nested sums, for these iteration equations for arbitrary $P$?
For example, for $P=2$ a closed form equation is:
$$A_{2,1,n+1} = 2 Z_n A_{2,1,n} + 1 \\
A_{2,m,n+1} = 2 Z_n A_{2,m,n} + \sum_{k=1}^{m-1} A_{2,k,n} A_{2,m-k,n}\quad (m \gt 1)$$
 A: I think the answer for arbitrary $P$ is "no".  It is possible to generate the series recurrence for arbitrary fixed $P$, but the recurrence contains nested sums with depth $P-1$, which is hard to express in a closed form (that can be translated to computer code without too much effort).  Some experiments with the Maxima computer algebra system:
$ maxima
(%i1) sumexpand: true$
(%i2) cauchysum: true$
(%i3) F(p, c, z) := z^p + c$
(%i4) D(p, c, z) := F(p, c + C, z + Z) - F(p, C, Z)$
(%i5) A(p) := niceindices(collectterms(expand(D(p,c,sum(a[i]*c^i,i,1,inf))),c))$
(%i6) tex(A(2))$

$$\sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+2\,Z
 \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$
(%i7) tex(A(3))$

$$\sum_{j=3}^{\infty }{c^{j}\,\sum_{{\it i_6}=1}^{j-2}{a_{{\it i_6}}
 \,\sum_{i=1}^{j-{\it i_6}-1}{a_{i}\,a_{j-{\it i_6}-i}}}}+3\,Z\,
 \sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+3\,Z^2
 \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$
(%i8) tex(A(4))$

$$\sum_{k=4}^{\infty }{c^{k}\,\sum_{{\it i_{14}}=1}^{k-3}{a_{
 {\it i_{14}}}\,\sum_{j=1}^{k-{\it i_{14}}-2}{a_{j}\,\sum_{i=1}^{k-j-
 {\it i_{14}}-1}{a_{i}\,a_{k-j-{\it i_{14}}-i}}}}}+4\,Z\,\sum_{j=3}^{
 \infty }{c^{j}\,\sum_{{\it i_{18}}=1}^{j-2}{a_{{\it i_{18}}}\,\sum_{
 i=1}^{j-{\it i_{18}}-1}{a_{i}\,a_{j-{\it i_{18}}-i}}}}+6\,Z^2\,
 \sum_{j=2}^{\infty }{c^{j}\,\sum_{i=1}^{j-1}{a_{i}\,a_{j-i}}}+4\,Z^3
 \,\sum_{i=1}^{\infty }{c^{i}\,a_{i}}+c$$
These are not quite the desired form (have to collect a few things to get the coefficient of $c^j$) but are close enough for now.  And it seems niceindices() fails... (Maxima 5.38.1 on Debian).
