[seqfan] A251690: Any Pattern to the Terms?

[Paul D Hanna]

SeqFans, 
     Please consider the sequence    http://oeis.org/A251690 
 
Can someone extend to more terms (or) analyze why the terms seem to remain in the interval [-3,0] for n>1? 
I wonder if there is a pattern to the terms ...   
 
A251690 describes the integer series A(x) such that G(A(x) - x^k) has negative coefficients for all k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764. 
  
I looked at this G(x) because I found that, for the Catalan function C(x) = 1 + x*C(x)^2, the integer series F(x), such that C(F(x)-x^k) has negative coefficients, is simple and generates the Fibonacci series C(F(x)) = 1/(1-x-x^2).   
 
Thanks, 
     Paul