0,5

This can be described in 2 ways: (a) Trees with n nodes of valency <= 3, for n = 0,1,2,3,... (b) Trees with t = 2n+2 nodes of valency either 1 or 3 (implying that there are n nodes of valency 3 - the boron atoms - and n+2 nodes of valency 1 - the hydrogen atoms), for t = 2,4,6,8,...

P. J. Cameron, Oligomorphic Permutation Groups, Cambridge; see Fig. 2 p. 35.

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).

S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261.

R. C. Read, personal communication.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

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

Eric Weisstein's World of Mathematics, Trivalent Tree

Rains and Sloane give a g.f.

a(0)=a(1)=a(2)=1, a(n) = 2b(n+1) - b(n+2) + b((n+1)/2) - 2 C(1+b(n/3), 3) - Sum_{i=1..[(n-1)/2]} C(b(i), 2)b(n-2i) + Sum_{i=1..[n/3]} b(i) Sum_{j=i..[(n-i)/2]} b(j)b(n-i-j), where b(x) = A001190(x) if x is an integer, otherwise 0 (Cyvin et al.)

The 4 trees with 6 nodes are:

._._._._._. . ._._._._. . ._._._._. . ._._._.

. . . . . . . . | . . . . . . | . . . . | |

Equals A000673 + A000675. Cf. A052120, A000022, A000200, A000602.

Sequence in context: A032237 A124346 A033961 this_sequence A115868 A103299 A010101

Adjacent sequences: A000669 A000670 A000671 this_sequence A000673 A000674 A000675

nonn,easy,nice

njas