A000621 G.F.: Conflicting Formulas

[Paul D. Hanna]

Seqfans, 
 
Someone wrote: 
> Since you are given as the reference for the (2), I wonder how you 
> came up with it. 
 
I was interested back then in documenting recursive equations 
that seem to occur in nature, and so perhaps I found that A000621 
agreed with 40 initial terms of the equation A(x) = 1/(1 - x*A(x^2)).  
In retrospect, I should have indicated "conjecture" next to the g.f. 
since even 40 initial terms is not enough to be certain. 
 
> (without seeing the original references, your gf seems 
> more plausible, because of the recursive nature of the original 
> combinatorial situation, but who knows :) )
 
Agreed - the recursive equation is most likely; an opinion supported by 
the g.f. for the closely related sequence A000625:  
G.f. satisfies A(x) = 1 + x*(A(x)^3 + 2*A(x^3))/3. 
 
As Neil suggested, I e-mailed Bruce Corrigan for possible clarification. 
Thanks to all for your comments.
 
Regards,
     Paul
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