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[Christian G. Bower]

ftaw
> I think we need to look at one more table:  the number of connected
> categories with n morphisms and k objects.  This starts:

>   1
>   2
>   7 1
>  35 6
> 228 28 2

> (Row n has length ceiling(n/2).)
...
> For the table above, a(2n-1,n) has a purely algebraic or graph-
> theoretic interpretation.  It is the number of connected
> anti-transitive relations on n objects (meaning that if a R b and
> b R c, then NOT (a R c)); equivalently, the number of bipartite
> oriented trees, where each edge origin is in the same part.
> 
> If I haven't made any mistakes, this sequence starts:
> 
> 1,1,2,3,6,10
> 
> This is not enough data to determine whether it is in the OEIS.

For n>1 it's A122086
http://www.research.att.com/~njas/sequences/A122086
A sequence which could use a nicer description and formula such as
a(n) = A000055(n) - A000081(n/2) where a sequence evaluated at a
noninteger is 0.
http://www.research.att.com/~njas/sequences/A000055
http://www.research.att.com/~njas/sequences/A000081

Christian