OFFSET
1,3
COMMENTS
The degree of each node is <= 4.
A bicentroid is an edge which connects two subtrees of exactly m/2 nodes, where m is the number of nodes in the tree. If a bicentroid exists it is unique. Clearly trees with an odd number of nodes cannot have a bicentroid.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A086200 for the analogous sequence with stereoisomers counted.
REFERENCES
F. Harary, Graph Theory, p. 36, for definition of bicentroid.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..500 (terms n = 1..100 from Vincenzo Librandi)
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
FORMULA
a(n) = b(n)*(b(n)+1)/2, where b(n) = A000598[ n ].
MAPLE
M[1146] := [ T, {T=Union(Epsilon, U), U=Prod(Z, Set(U, card<=3))}, unlabeled ]:
bicenteredHC := proc(n) option remember; if n mod 2<>0 then 0 else binomial(count(M[ 1146 ], size=n/2)+1, 2) fi end:
MATHEMATICA
m = 24; a[x_] = Sum[c[k]*x^k, {k, 0, m}]; s[x_] = Series[ 1 + (1/6)*x*(a[x]^3 + 3*a[x]*a[x^2] + 2*a[x^3]) - a[x], {x, 0, m}]; eq = Thread[ CoefficientList[s[x], x] == 0];
Do[so[k] = Solve[eq[[1]], c[k-1]][[1]]; eq = Rest[eq] /. so[k], {k, 1, m+1}]; b = Array[c, m, 0] /. Flatten[ Array[so, m+1] ]; Rest[b*(b+1)/2] (* Jean-François Alcover, Jul 25 2011, after A000598 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
STATUS
approved