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%I #39 Oct 27 2023 21:08:16
%S 1,1,3,10,36,153,780,4005,22366,128778,766941,4674153,29180980,
%T 185117661,1193918545,7800816871,51584238201,344632209090,
%U 2324190638055,15804057614995,108277583483391,746878494484128,5183852459907628
%N Number of unrooted quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n+2) with a bicentroid, ignoring stereoisomers.
%C The degree of each node is <= 4.
%C 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.
%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 A086200 for the analogous sequence with stereoisomers counted.
%D F. Harary, Graph Theory, p. 36, for definition of bicentroid.
%H Alois P. Heinz, <a href="/A010373/b010373.txt">Table of n, a(n) for n = 1..500</a> (terms n = 1..100 from Vincenzo Librandi)
%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>
%F a(n) = b(n)*(b(n)+1)/2, where b(n) = A000598[ n ].
%p M[1146] := [ T,{T=Union(Epsilon,U),U=Prod(Z,Set(U,card<=3))},unlabeled ]:
%p bicenteredHC := proc(n) option remember; if n mod 2<>0 then 0 else binomial(count(M[ 1146 ],size=n/2)+1,2) fi end:
%t m = 24; a[x_] = Sum[c[k]*x^k, {k, 0, m}]; s[x_] = Series[ 1 + (1/6)*x*(a[x]^3 + 3*a[x]*a[x^2] + 2*a[x^3]) - a[x], {x, 0, m}]; eq = Thread[ CoefficientList[s[x], x] == 0];
%t Do[so[k] = Solve[eq[[1]], c[k-1]][[1]]; eq = Rest[eq] /. so[k], {k, 1, m+1}]; b = Array[c, m, 0] /. Flatten[ Array[so, m+1] ]; Rest[b*(b+1)/2] (* _Jean-François Alcover_, Jul 25 2011, after A000598 *)
%Y A000602(n) = A010372(n) + a(n/2) for n even, A000602(n) = A010372(n) for n odd.
%Y Cf. A000200, A000598.
%K nonn,easy
%O 1,3
%A _Paul Zimmermann_, _N. J. A. Sloane_
%E Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
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