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

[pauldhanna at juno.com]

Seqfans,
     There is a simple answer my own questions (in prior email). 
Since asinh(x) = log(sqrt(1+x^2) + x), 
then the sum (13) fits the form:
  Sum_{n>=0} log(F(q^n*x))^n/n!  =  Sum_{n>=0} x^n [y^n] F(y)^(q^n)
where F(y) = sqrt(1+y^2) + y  .
 
Thus, 
G(x,q) = Sum_{n>=0} asinh( q^n*x )^n / n!  = 
  Sum_{n>=0} x^n [y^n] ( sqrt(1+y^2) + y )^(q^n) 
 
When q=2,
G(x,2) = Sum_{n>=0} asinh( 2^n*x )^n / n!  = 
  Sum_{n>=0} x^n [y^n] ( sqrt(1+y^2) + y )^(2^n) = 
  1 + 2*x + 8*x^2 + 84*x^3 + 2688*x^4 + 276892*x^5 + 94978048*x^6 +...
  
then to get the sum of (12) we simply take the bisection: (G(x,2) -
G(-x,2))/2.  
 
Thanks, 
      Paul 
  
Here is an example using the hyperbolic sine series applied on the
inverse sinh:
(12) G.f.: A(x) = Sum_{n>=0} asinh( 2^(2n+1)*x )^(2n+1) / (2n+1)!  = 
2*x + 84*x^3 + 276892*x^5 + 111457917800*x^7 + 
6660816097416169260*x^9 + 66597307693046550483175282456*x^11 +
120167520447600665027319450022840022638104*x^13 +...

Is there a simple formula for the integer coefficients on the right side
of (12)? 
 

Of course (12) is an example of the more general: 
(13) G.f.: A(x) = Sum_{n>=0} asinh( q^(2n+1)*x )^(2n+1) / (2n+1)!  = ? 
yielding some unknown integer series for all integer q. 

Is there a simple formula for the integer coefficients on the right side
of (13)? 
It would be nice if it turned out to be as simple a formula as: 
(2) Sum_{n>=0} log(1 + q^n*x)^n/n!  =  Sum_{n>=0} C(q^n,n)*x^n. 
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