[seqfan] Series Satisfying Certain Functional Equations

[Paul D Hanna]

SeqFans, 
   Consider the family of power series A(x) and G(x) defined by the following. 
 
For a fixed positive integer n, define an integer series A(x) such that: 
  
  A(x)^(2^(n+1)) = A(x^2)^(2^n) + 2^(n+1)*x 
  
then there exists an integer series G(x) that satisfies: 
  
  A(x)^(2^n)  =  1/G(x)^(2^n) + 4^n*x*G(x)^(2^n)  and 
 
  A(x)^(2^n)  =  1/G(x^2)^(2^(n-1)) + 2^n*x*G(x^2)^(2^(n-1)). 
 
Please see the examples shown below.    
  
Q: What other formulas can be given for the g.f.s A(x) and G(x)? 
 
I'd also like to know any asymptotics or limits that may exist. 
 
Best Regards, 
   Paul 
 
P.s.: I am in the process of formatting these sequences for submission, 
so not all are yet in the OEIS. 
 
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EXAMPLES. 
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CASE n=1. 
 
Let A(x) and G(x) satisfy  
  A(x)^4 = A(x^2)^2 + 4*x 
and  
  A(x)^2 = 1/G(x)^2 + 4*x*G(x)^2 = 1/G(x^2) + 2*x*G(x^2).   
Coefficients begin   
A: [1, 1, -1, 2, -5, 13, -35, 99, -289, 857, -2578, 7864, ...];
G: [1, 1, 6, 41, 334, 2901, 26651, 253709, 2483395, 24829132, ...].
  
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CASE n=2. 
 
Let A(x) and G(x) satisfy  
  A(x)^8 = A(x^2)^4 + 8*x 
and   
  A(x)^4 = 1/G(x)^4 + 16*x*G(x)^4 = 1/G(x^2)^2 + 4*x*G(x^2)^2. 
 
Coefficients begin   
A: [1, 1, -3, 14, -76, 441, -2678, 16813, -108093, 707451, ...];
G: [1, 3, 72, 2307, 86295, 3513477, 151235361, 6768437853, ...].
 
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CASE n=3.  
 
Let A(x) and G(x) satisfy  
  A(x)^16 = A(x^2)^8 + 16*x 
and  
  A(x)^8 = 1/G(x)^8 + 64*x*G(x)^8 = 1/G(x^2)^4 + 8*x*G(x^2)^4. 
 
Coefficients begin   
A: [1, 1, -7, 70, -798, 9737, -124124, 1631041, -21911758, ...];
G: [1, 7, 672, 91147, 14486409, 2516759469, 463051052653, ...].
 
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CASE n=4.  
 
Let A(x) and G(x) satisfy  
  A(x)^32 = A(x^2)^16 + 32*x 
and  
  A(x)^16 = 1/G(x)^16 + 256*x*G(x)^16 = 1/G(x^2)^8 + 16*x*G(x^2)^8. 
 
Coefficients begin    
A: [1, 1, -15, 310, -7250, 181257, -4727688, 126967785, ...];
G: [1, 15, 5760, 3203115, 2094795645, 1500292338765, ...].
 
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CASE n=5. 
  
Let A(x) and G(x) satisfy   
  A(x)^64 = A(x^2)^32 + 64*x 
and  
  A(x)^32 = 1/G(x)^32 + 1024*x*G(x)^32 = 1/G(x^2)^16 + 32*x*G(x^2)^16.   
 
Coefficients begin   
A: [1, 1, -31, 1302, -61690, 3121545, -164661088, ...];
G: [1, 31, 47616, 107151531, 284175325349, 826107706608333, ...].
 
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END.