More Functional Puzzles

[Paul D. Hanna]

Seqfans, 
     I wish to share a few more "functional puzzles" with solutions. 
Slight variations could produce more interesting results. 
     Paul 
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Consider the systems of simultaneous equations below; for each case, 
what is the unique solution to variable A as a formal power series in x? 
 
(1)
A = 1 + xAB
B = A + xBC
C = B + xCD
D = C + xDE
E = D + xEF
...
   
(2a) 
A = (1 + xB)
B = A(1 + xC)
C = B(1 + xD)
D = C(1 + xE)
E = D(1 + xF)
... 
(2b)  
A = 1 + xAB
B = 1 + xABC
C = 1 + xABCD
D = 1 + xABCDE
E = 1 + xABCDEF 
...
(same solution as (2a)).
 
(3) 
A = 1 + xB
B = 1 + xAC
C = 1 + xABD
D = 1 + xABCE
E = 1 + xABCDF
...
     
SOLUTIONS.
(1) A satisfies:  A(x) = 1 + x*A(x)^2*A(xA(x)) = g.f. of A088714: 
[1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825,...].  
  
(2a) and (2b): A satisfies:  A(x) = 1 + xA(x)*A(xA(x)) 
where xA(x) = g.f. of A030266, and coefficients of A begin: 
[1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764,...]. 
 
(3) A satisfies:  A(x) = 1 + xA(xA(x)) = g.f. of A087949: 
[1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428, 174835,...]. 
 
END.
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