G.f. for C(q^n,n)?

[pauldhanna at juno.com]

Seqfans, 
        Just found an o.g.f. for A014070(n) = C(2^n,n): 
 
(1) G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!.
  
More generally, we have the nontrivial identity: 
 
(2) Sum_{n>=0} log(1 + q^n*x)^n/n!  =  Sum_{n>=0} C(q^n,n)*x^n. 
 
The proof is easy given the following formula for A008275, 
the Stirling numbers of the first kind: 
   E.g.f. for k-th column of A008275 is ln(1+x)^k/k!. 
 
My questions: 
Can any other interesting result be derived from (2)? 
Thanks, 
     Paul 
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