A086200 Number of unrooted steric quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n +2) with a bicentroid when stereoisomers are regarded as different.

1, 3, 15, 66, 406, 2775, 19900, 152076, 1206681, 9841266, 82336528, 702993756, 6105180250, 53822344278, 480681790786, 4342078862605, 39621836138886, 364831810979041, 3386667673687950, 31669036266203766

Offset: 1

Author

Steve Strand (snstrand(AT)comcast.net), Aug 28 2003

Keywords

• nonn

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.

Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A010373 for the analogous sequence when stereoisomers are not counted as different.

Links

• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000598, A000602, A010732, A010733, A000625, A000628, A086194.

For even n A000628(n) = A086194(n) + a(n/2), for odd n A000628(n) = A086194(n), since every tree has either a centroid or a bicentroid but not both.

Formula

G.f.: replace each term x in g.f. for A000625 by x(x+1)/2. Interpretation: ways to pick 2 specific radicals (order not important) from all n carbon radicals is number of 2n carbon bicentered alkanes (join the two radicals with an edge).