Functional Puzzle

[Paul D. Hanna]

Seqfans, 
      I don't think I was very clear. Perhaps this is better: 
A(x) = 1 + x*A(x)*B(x)
B(x) = 1 + x*A(x)*B(x)*C(x)
C(x) = 1 + x*A(x)*B(x)*C(x)*D(x)
D(x) = 1 + x*A(x)*B(x)*C(x)*D(x)*E(x)
E(x) = 1 + x*A(x)*B(x)*C(x)*D(x)*E(x)*F(x)
...
This leads to the following products: 
A = 1/(1 - xB) 
AB = 1/(1 - xB - xC) 
ABC = 1/(1 - xB - xC - xD) 
ABCD = 1/(1 - xB - xC - xD - xE) 
ABCDE = 1/(1 - xB - xC - xD - xE - xF) 
...
Or, equivalently,  
A = 1 + xB/(1 - xB) 
B = 1 + xC/(1 - xB - xC) 
C = 1 + xD/(1 - xB - xC - xD) 
D = 1 + xE/(1 - xB - xC - xD - xE) 
E = 1 + xF/(1 - xB - xC - xD - xE - xF) 
...
  
The final relation gives me the coefficients for A(x) to be: 

1, 1, 2, 6, 22, 93, 439, 2269, 12672, 75751, 481277, 3232126, 22843829, 
169308433, 1311923063, 10600748257, 89122102396, 778020244975, 
7040120987869, 65925795526266, 637943017576192, 6370573324549624, ...
  
So, sorry, my initial calculation was incorrect!  
The sequence is not found in the OEIS after all. 
But I thought the problem interesting anyway. 
  
Thanks, 
     Paul 
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