|
%I #24 Dec 19 2020 03:12:56
%S 1,0,1,1,3,2,9,8,35,39,159,202,802,1078,4347,6354,24894,38157,148284,
%T 237541,910726,1511717,5731580,9816092,36797588,64658432,240215803,
%U 431987953,1590507121,2917928218,10660307791,19910436898
%N 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.
%C The degree of each node is <= 4.
%C 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.
%C 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.
%D F. Harary, Graph Theory, p. 36, for definition of centroid.
%H A. Cayley, <a href="/A000022/a000022.pdf">Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen</a>, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
%H E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p with(combstruct): Alkyl := proc(n) combstruct[count]([ U,{U=Prod(Z,Set(U,card<=3))},unlabeled ],size=n) end:
%p 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));
%p 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);
%p 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);
%Y Cf. A010373, A000022, A086194, A000598, A000602.
%Y A000602(n) = a(n) + A010373(n/2) for n even, A000602(n) = a(n) for n odd.
%K nonn,easy,nice
%O 1,5
%A _Paul Zimmermann_, _N. J. A. Sloane_
%E Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
|
