This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102911 #17 Aug 11 2025 06:34:36 %S A102911 0,1,1,3,10,45,210,1176,6670,41041,258840,1697403,11359761,77956341, %T A102911 543625851,3855429766,27702225271,201515674128,1481195012220, %U A102911 10991843660826,82256068767106,620288742329028,4709854127998971,35987845277616940,276563426284762620 %N A102911 Number of unlabeled (and unrooted) trees on 2n nodes with a bicentroid. %C A102911 A tree has either a center or a bicenter and either a centroid or a bicentroid. (These terms were introduced by Jordan.) %C A102911 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 A102911 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 A102911 A 2n-node tree with a bicentroid consists of two n-node rooted trees with the roots joined by an edge. %D A102911 F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1994; pp. 35, 36. %H A102911 N. J. A. Sloane, <a href="/A102911/b102911.txt">Table of n, a(n) for n = 0..200</a> %H A102911 A. Cayley, <a href="http://www.jstor.org/stable/2369158">On the analytical forms called trees</a>, Amer. J. Math., 4 (1881), 266-268. %H A102911 C. Jordan, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0070">Sur les assemblages des lignes</a>, J. Reine angew. Math., 70 (1869), 185-190. %H A102911 E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.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.] %F A102911 a(n) = r(n)*(r(n)+1)/2 where r(n) = A000081(n) is the number of rooted trees on n nodes. %F A102911 Let f(n) = a(n/2) if n is even, = 0 otherwise. Then f(n) + A027416(n) = A000055(n). %Y A102911 Cf. A027416 (trees with a centroid), A000676 (trees with a center), A000677 (trees with a bicenter), A000055 (trees), A000081 (rooted trees). %K A102911 nonn %O A102911 0,4 %A A102911 _N. J. A. Sloane_ and _David Applegate_, Feb 26 2007