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A010372
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Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) with a centroid ignoring stereoisomers.
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7
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1, 0, 1, 1, 3, 2, 9, 8, 35, 39, 159, 202, 802, 1078, 4347, 6354, 24894, 38157, 148284, 237541, 910726, 1511717, 5731580, 9816092, 36797588, 64658432, 240215803, 431987953, 1590507121, 2917928218, 10660307791, 19910436898
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OFFSET
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1,5
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COMMENTS
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The degree of each node is <= 4.
A centroid is a node with less than n/2 nodes in each of the incident subtrees, where n is the number of nodes in the tree. If a centroid exists it is unique.
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 A086194 for the analogous sequence with stereoisomers counted.
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REFERENCES
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F. Harary, Graph Theory, p. 36, for definition of centroid.
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LINKS
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Table of n, a(n) for n=1..32.
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.
Index entries for sequences related to trees
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MAPLE
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with(combstruct): Alkyl := proc(n) combstruct[count]([ U, {U=Prod(Z, Set(U, card<=3))}, unlabeled ], size=n) end:
centeredHC := proc(n) option remember; local f, k, z, f2, f3, f4; f := 1 + add(Alkyl(k)*z^k, k=0..iquo(n-1, 2));
f2 := series(subs(z=z^2, f), z, n+1); f3 := series(subs(z=z^3, f), z, n+1); f4 := series(subs(z=z^4, f), z, n+1);
f := series(f*f3/3+f4/4+f2^2/8+f2*f^2/4+f^4/24, z, n+1); coeff(f, z, n-1) end: seq(centeredHC(n), n=1..32);
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CROSSREFS
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Cf. A010373, A000022, A086194, A000598, A000602.
A000602(n) = a(n) + A010373(n/2) for n even, A000602(n) = a(n) for n odd.
Sequence in context: A081233 A050676 A356185 * A199455 A287768 A197831
Adjacent sequences: A010369 A010370 A010371 * A010373 A010374 A010375
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Paul Zimmermann, N. J. A. Sloane
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EXTENSIONS
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Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
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STATUS
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approved
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