Sequences A120588 - A120607 : Formulas?

[Paul D. Hanna]

Seqfans,
      Thanks, Ralf and Emeric, for your interesting thoughts ; 
however, there is a difference between your g.f.s 
and the g.f.s that I am interested in here.
 
The point of interest (to me) in the g.f. A(x) that satisfies: 
(1)   r*A(x) = c + b*x + A(x)^n 
is that A(x) is generated from A(x)^n without "shifting" of terms.  
 
Let a_n(k) be the k-th term of A(x)^n where A(x) satisfies (1), 
then 
   a_n(k) / a(k)  =  r   for all k>1. 
One would expect that this simple condition would likely ensure 
an explicit formula for the terms a(k). 
 
SERIES SOLUTION. 
Well, the following is not explicit, but one can apply the 
Lagrange Inversion Theorem 
http://mathworld.wolfram.com/LagrangeInversionTheorem.html
to obtain an infinite series solution to (1): 
  
(2) A(x) = Sum_{i>=0} C(n*i,i)/(n*i-i+1)*(c+bx)^(n*i-i+1)/r ^(n*i+1) 
 
given c, b, and r are sufficient to allow convergence. 
 
EXAMPLE.  
Take the simplest case: 
g.f. A(x) of A120588 satisfies: 3*A(x) = (2 + x) + A(x)^2. 
Then 
A(x) = Sum_{n>=0} C(2n,n)/(n+1)*(2+x)^(n+1) / 3^(2*n+1) 
or:
A(x) = Sum_{n>=0} A000108(n) * (2+x)^(n+1) / 3^(2*n+1) 
      = Catalan( (2+x)/9 )*(2+x)/3  
      = 1 + x*Catalan(x) 
which is indeed the solution. 
END_EXAMPLE.
 
EXPLICIT FORMULA? 
So the question now is: 
can the infinite series expression (2) be simplified any further 
to obtain an explicit formula for a(n)?  
 
Thanks,
      Paul 



Dear seqfan friends,
The recent confusion with A000446 and A016032 is just one of
the many opportunities I have to point out that WE ALL
Of course, definitions should be precise. In addition,
examples should be given because, in case of need, they also
help us to understand the definition.
Also, terms used in the definition, that are not generally
known should be defined.
After all, an accepted new sequence is a short published
paper, which was not subjected to the standard refereeing
process. Very often, I think most of the time, papers
returned to the author for revision (typos, errors, mode of
presentation, spelling, etc.). In the case of the OEIS
revision.
Best regards,
Emeric Deutsch
P.S. I am aware that this message reaches X "innocents" 
and misses Y "culprits". 

Sorry, if some of you are receiving this for the 2nd time.