Functional Puzzle

[Paul D. Hanna]

Seqfans, 
     Here is nice problem that may interest you: 
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Find the unique integer function A(x) that satisfies: 
A = 1 + xAB 
B = 1 + xABC 
C = 1 + xABCD 
D = 1 + xABCDE 
E = 1 + xABCDEF 
F = 1 + xABCDEFG 
...
The above nested relation continues on to infinity, 
where B(x), C(x), D(x), ... are also integer functions. 
-------------------------------------------------- 
 
If my answer is correct, the solution to A is quite unexpected, 
and is in the OEIS as one of Christian Bower's nice sequences. 
But I would like someone to confirm my results. 
 
The complete answer would make for a nice rectangular table 
where the coefficients of each function A,B,C,... form the rows. 
 
Thanks, 
      Paul 
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