A111814: Odd E.G.F.

[Paul D. Hanna]

Seqfans,
        Consider the peculiar generating function:
  
(*)  x/(1-x) = Sum_{n>=1} (1/n!)*Product_{j=0..n-1} A(2^j*x)
e.g.,
x/(1-x) = A(x) + A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3!
           + A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! 
           + A(x)*A(2*x)*A(2^2*x)*A(2^3*x)*A(2^4*x)/5! + ...
  
A solution, A(x), is the e.g.f. of sequence A111814:
A(x) = x - 2*x^3/3! + 216*x^5/5! - 568464*x^7/7! + 36058658688*x^9/9! 
              - 53694310935340800*x^11/11! +-...
  
It appears that A(x) is an odd function, but for no compelling reason.
 
Can anyone shed light on the mystery as to 
why A(-x) = -A(x) would be a necessary consequence of (*)?
 
The function A(x) is very interesting since the following function G(x):
(**)  G(x) = Sum_{n>=1} (m*2^k)^n/n!*Product_{j=0..n-1} A(2^j*x)
is always an integer series for all integer m and integer k>=0, and
G(x) equals the g.f. of column k of the matrix m-th power of triangle
A078121.
This (**) is a result of an observation made by Gottfried Helms over 2
years ago! 
 
Thanks,
      Paul
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