What is A007835?

[Brendan McKay]

Can't help you with the meaning of this sequence, but I disagree
with you on terminology.  In the graph theory literature, "directed
trees" usually have all their edges oriented towards a single
vertices, or all oriented away from a single vertex.  I think this
is a lot more common than using "oriented tree" for this meaning.
I would call a tree with edges oriented in arbitrary directions
an "oriented tree" as that is consistent with the very common
"oriented graph" - but of course a few words of clarification
can never hurt.

Brendan.

* franktaw at netscape.net <franktaw at netscape.net> [060518 09:39]:
> The entry for this sequence is:
>  
> %I A007835
> %S A007835 1,1,3,8,21,52,124,284,629,1352,2829
> %N A007835 Number of unordered sets of pairs (in-degree, out-degree) for nodes of oriented  trees.
> %H A007835 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
> %Y A007835 Cf. A000238.
> %K A007835 nonn,more
> %O A007835 1,3
> %A A007835 Philippe Aubry (aubry(AT)xlstat.com)
> 
> I have not been able to make sense out of this.
>  
> I was able to determine that by "oriented tree" is meant a directed graph (digraph) that, when the directions of the edges are erased, is a (free) tree.  (An unfortunate bit of nomenclature, that, since "oriented tree" is also used as a synonym for "rooted tree" - such as Knuth, The Art of Computer Programming.  "Directed tree" would have been a much better choice.)
>  
> However that may be, there are 3 of these "oriented trees" with 3 nodes:
>  
> O-->O-->O, O-->O<--O, and O<--O-->O.
>  
> The first of these has (in-degree, out-degree) pairs (0,1), (1,1), and (1,0); the second has (0,1) twice and (2,0), and the third has (1,0) twice and (0,2).  If we simply take the set of all such unordered pairs, that would have 3 elements: {(0,1), (0,2), (1,1)}.  However, then for n=4, we get only {(0,1), (0,2), (1,1), (0,3), (1,2)}, for 5, not 8.  (And in general, we would get A024206.)  I don't see any reasonable way to get a larger value for a(4) that does not make a(3) be at least 4.
>  
> Franklin T. Adams-Watters
> ___________________________________________________
> Try the New Netscape Mail Today!
> Virtually Spam-Free | More Storage | Import Your Contact List
> http://mail.netscape.com