A055779

[franktaw at netscape.net]

I am unable to make sense of this sequence.  The definition of "fat 
tree" reads: "A fat tree on vertex set V is a partition of V together 
with edges that link the parts of the partition in a tree-like pattern: 
that is, when the parts are collapsed to points, the edges are a tree." 
 From offset 1, the first 3 values given are 1,2,10.

This definition doesn't specify whether the vertices are to be 
considered distinct (although from context it appears that they are), 
nor what kind of tree is meant (the context suggests free trees).  
However, none of the available choices produce these numbers.

Distinct vertices, free trees: 1,2,7.
Distinct vertices, rooted trees: 1,3,16.
Identical vertices, free trees: 1,2,3.
Identical vertices, rooted trees: 1,2,5.

There is a formula given, in Tex, but I can't make sense of it either: 
"$\frac{n!}{n^2} Sum_{\mu \vdash n} \prod_{j=1}^{\infty} 
\frac{n^{\mu_j}}{{\mu_j}!(j-1)!^{\mu_j}}$".  In the sum, it appears 
that mu is a simple integer, but inside the product it is treated as a 
sequence.

Franklin T. Adams-Watters

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