Conjugate m-dimensional partitions

[A.N.W.Hone at kent.ac.uk]

Not so unlikely! The two conditions are completely equivalent. 

Condition (2) can be rewritten as 

x*A(x)=f^(-1)(x), 

where f(x)=x*A(-x). Applying f to both sides gives 

f(x*A(x))=x 

or 

x*A(x)*A(-x*A(x))=x, 

which immediately gives condition (1): 

A(x) = 1/A(-x*A(x)).

Actually, substituting a general series A(x)=a_0 + a_1*x + a_2*x^2 +... 
into either functional equation (1) or (2), one finds that the even terms are 
uniquely determined at each order by the previous ones, but the odd terms are 
arbitrary. So any formal series of the form  

A(x)=1 + a_1*x + a_1^2*x^2 + a_3*x^3  + (3*a_1*a_3-2*a_1^4)*x^4 + a_5*x^5 
+  (4*a_1*a_5+2*a_3^2-18*a_1^3*a_3+13*a_1^6)*x^6 + a_7*x^7 + ... 

satsifies the functional equation. 

The case you are interested in corresponds to a_1=1, a_3=2, a_5=7, a_7=24,... 
so some other consistent functional relation would be needed to uniquely 
determine the particular series 

1+x+x^2+2*x^3+4*x^4+7*x^5+13*x^6+24*x^7+42*x^8+... 

Andy 
 

----- Original Message -----
From: franktaw at netscape.net
Date: Tuesday, May 9, 2006 7:24 pm
Subject: Re: Conjugate m-dimensional partitions
To: seqfan at ext.jussieu.fr

> It does seem unlikely.
>  
> Is there a unique sequence that satisfies these two 
> conditions?  If so, I would think it a worthwhile thing to 
> add to the OEIS.  (If it's not there already - it might be 
> A002547.) 
> Franklin T. Adams-Watters
> 
> -----Original Message-----
> From: Paul D Hanna pauldhanna at juno.com
> 
> 
> Franklin, 
>      BTW, should *both* of those conditions 
> continue to hold 
> (I would be surprised),  the next term is required to 
> be:  42.
>  
> Paul
>  
> -- "Paul D Hanna" <pauldhanna at juno.com> wrote:
> Franklin (and seqfans),
>     Perhaps this is just coincidence, 
> but so far the g.f. A(x) of the terms
> > 1,1,1,2,4,7,13,24 
>  
> satisfies these 2 relations: 
>  
> (1) A(x) = 1/A(-x*A(x))
>   
> (2) x*A(x) = series_reversion(x*A(-x))
>  
> It would be curious (since unlikely) if these trends continue.
> 
> Paul
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