%I A027416
%S A027416 1,1,0,1,1,3,3,11,13,47,61,235,341,1301,1983,7741,12650,48629,82826,
%T A027416 317955,564225,2144505,3926353,14828074,27940136,104636890,201837109,
%U A027416 751065460,1479817181,5469566585,10975442036,40330829030,82270184950
%N A027416 Number of unlabeled (and unrooted) trees on n nodes having a centroid.
%C A027416 Also, number of rooted unlabeled trees on n nodes not having a primary 
               branch.
%C A027416 A tree has either a center or a bicenter and either a centroid or a bicentroid. 
               (These terms were introduced by Jordan.)
%C A027416 If the number of edges in a longest path in the tree is 2m, then the 
               middle node in the path is the unique center, otherwise the two middle 
               nodes in the path are the unique bicenters.
%C A027416 On the other hand, define the weight of a node P to be the greatest number 
               of nodes in any subtree connected to P. Then either there is a unique 
               node of minimal weight, the centroid of the tree, or there is a unique 
               pair of minimal weight nodes, the bicentroids.
%C A027416 Let T be a tree with root node R. If R and the edges incident with it 
               are deleted, the resulting rooted trees are called branches. A primary 
               branch (there can be at most one) has i nodes where n/2 <= i <= n-1.
%D A027416 A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 
               266-268.
%D A027416 F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36.
%D A027416 C. Jordan, Sur les assemblages des lignes, J. Reine angew. Math., 70 
               (1869), 185-190.
%D A027416 A Meir and J. W Moon, On the branch-sizes of rooted unlabeled trees, 
               in "Graph Theory and Its Applications", Annals New York Acad. Sci., 
               Vol. 576, 1989, pp. 399-407.
%H A027416 N. J. A. Sloane, <a href="b027416.txt">Table of n, a(n) for n = 0..200</
               a>
%H A027416 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1. 
               [This articles states incorrectly that A000676 and A000677 give the 
               numbers of trees with respectively a centroid and bicentroid.]
%H A027416 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A027416 a(n)=A000055(n) - A102911(n).
%F A027416 a(n)=A000081(n) - A027415(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 21 2004
%p A027416 N := 50: Y := [ 1,1 ]: for n from 3 to N do x*mul( (1-x^i)^(-Y[ i ]), 
               i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); Y := [ op(Y),b ]; 
               od: P:=n->sum(Y[n-i]*Y[i],i=1..floor(n/2)): seq(Y[n]-P(n),n=1..35); 
               (Deutsch)
%Y A027416 Cf. A102911 (trees with a bicentroid), A027415 (trees without a primary 
               branch), A000676 (trees with a center), A000677 (trees with a bicenter), 
               A000055 (trees), A000081 (rooted trees).
%Y A027416 Sequence in context: A136123 A045495 A045494 this_sequence A163932 A007022 
               A011950
%Y A027416 Adjacent sequences: A027413 A027414 A027415 this_sequence A027417 A027418 
               A027419
%K A027416 nonn
%O A027416 0,6
%A A027416 N. J. A. Sloane (njas(AT)research.att.com).
%E A027416 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 21 2004
%E A027416 Entry revised (with new definition) by N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 26 2007