[seqfan] G.f. for trees with degree at most 3

[franktaw at netscape.net]

I asked the author of this sequence for clarification, but got no 
response. Maybe somebody here can figure out what is going on.

http://www.research.att.com/~njas/sequences/A003692 
 
For this sequence, a generating function is given:

(1-x)(2-x-x^2) - (2-x+x^2)\sqrt{1-2x-x^2} \over 3 x^3. 
 
I'm not sure if this is supposed to be 
 
(1-x)*(2-x-x^2)-(2-x-x^2)*sqrt(1-2*x-x^2)/(3*x^3) 
 
or 
 
((1-x)*(2-x-x^2)-(2-x-x^2)*sqrt(1-2*x-x^2))/(3*x^3), 
 
but either one produces a series that includes terms with negative 
exponents; and in neither case there is any apparent relationship to 
this sequence. So what is the correct g.f.? 
 
Franklin T. Adams-Watters