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A086200
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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.
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5
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1, 3, 15, 66, 406, 2775, 19900, 152076, 1206681, 9841266, 82336528, 702993756, 6105180250, 53822344278, 480681790786, 4342078862605, 39621836138886, 364831810979041, 3386667673687950, 31669036266203766
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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LINKS
| Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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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).
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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.
Sequence in context: A001447 A106732 A052981 * A122558 A110211 A167874
Adjacent sequences: A086197 A086198 A086199 * A086201 A086202 A086203
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KEYWORD
| nonn
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AUTHOR
| Steve Strand (snstrand(AT)comcast.net), Aug 28 2003
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