[seqfan] A Remarkable Eigenfunction - A179497

[Paul D Hanna]

SeqFans, 
     
Consider the eigenfunction described by:
 
   E.g.f. satisfies: A(A(x))^2 = A(x)^2 * A'(x) 
  
   [http://www2.research.att.com/~njas/sequences/A179497].
  
Explicitly, the series begins: 
 
   A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! + 297000*x^6/6! + 13705776*x^7/7! + 776778688*x^8/8! + 52511234688*x^9/9! + 4143702216960*x^10/10! +..
 
 
Let A_n(x) denote the n-th iteration of A(x) where A_{n+1}(x) = A_n(A(x)) with A_0(x)=x;
 
then the pattern continues: 
 
   [A_3(x)]^2 = A(x)^2 * A_2'(x), 
  
   [A_4(x)]^2 = A(x)^2 * A_3'(x),
 
   [A_5(x)]^2 = A(x)^2 * A_4'(x), ...
 
so that the iterations of A(x) satisfy: 
 
(*) [A_{n+1}(x)]^2 = A(x)^2 * A_n'(x)   for all n. 
 
 
I find formula (*) to be remarkable, even though the proof for n>0 
is an easy exercise in induction and the chain rule. 
 
 
Surely A(x) must have some other nice properties. 
 
What else can be said about A(x)? 
 
   Paul