%I A010372
%S A010372 1,0,1,1,3,2,9,8,35,39,159,202,802,1078,4347,6354,24894,38157,
%T A010372 148284,237541,910726,1511717,5731580,9816092,36797588,64658432,
%U A010372 240215803,431987953,1590507121,2917928218,10660307791,19910436898
%N 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.
%C A010372 The degree of each node is <= 4.
%C A010372 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 A010372 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 A010372 F. Harary, Graph Theory, p. 36, for definition of centroid.
%H A010372 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A010372 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%p A010372 with(combstruct): Alkyl := proc(n) combstruct[count]([ U,{U=Prod(Z,Set(U,
               card<=3))},unlabeled ],size=n) end:
%p A010372 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 A010372 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 A010372 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 A010372 Cf. A010373, A000022, A086194, A000598, A000602.
%Y A010372 A000602(n) = a(n) + A010373(n/2) for n even, A000602(n) = a(n) for n 
               odd.
%Y A010372 Sequence in context: A118045 A081233 A050676 this_sequence A152049 A099887 
               A038220
%Y A010372 Adjacent sequences: A010369 A010370 A010371 this_sequence A010373 A010374 
               A010375
%K A010372 nonn,easy,nice
%O A010372 1,5
%A A010372 Paul.Zimmermann(AT)inria.fr, N. J. A. Sloane (njas(AT)research.att.com).
%E A010372 Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 
               2003.