A035088 Number of labeled polygonal cacti (Husimi graphs) with n nodes.

1, 1, 0, 1, 3, 27, 240, 2985, 42840, 731745, 14243040, 313570845, 7683984000, 207685374435, 6135743053440, 196754537704725, 6805907485977600, 252620143716765825, 10015402456976716800, 422410127508300756825, 18884777200534941696000

Offset: 0

Author

Christian G. Bower, Nov 15 1998

Keywords

• nonn
• nice

Comments

A Husimi tree is a connected graph in which no line lies on more than one cycle [Harary, 1953]. - Jonathan Vos Post, Mar 12 2010

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
• F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141.
• F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
• F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39. pp. 315-322, 1953.
• F. Harary, G. Uhlenbeck (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences 39: 315-322. - From Jonathan Vos Post, Mar 12 2010

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Index entries for sequences related to cacti
• Index entries for sequences related to trees

Crossrefs

Cf. A035082, A035083, A035084, A035085, A035086, A035087.

Programs

Mathematica

max = 20; s = 1+InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, max}], x]; a[n_] := SeriesCoefficient[s, n]*(n-1)!; a[0]=1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 27 2016, after Vaclav Kotesovec at A035087 *)

Formula

a(n) = A035087(n)/n, n > 0.