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A010372 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. 7
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 (list; graph; refs; listen; history; text; internal format)



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.


F. Harary, Graph Theory, p. 36, for definition of centroid.


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


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);


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: A288055 A081233 A050676 * A199455 A287768 A197831

Adjacent sequences:  A010369 A010370 A010371 * A010373 A010374 A010375




Paul Zimmermann, N. J. A. Sloane


Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003



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Last modified December 11 07:18 EST 2019. Contains 329914 sequences. (Running on oeis4.)