A035086 Number of increasing rooted polygonal cacti (Husimi graphs) with n nodes.

1, 0, 1, 3, 19, 135, 1204, 12537, 150556, 2043930, 30969211, 517973148, 9478800604, 188381470095, 4040440921699, 93020386382742, 2287969523647171, 59877222907995675, 1661259526266784171, 48705364034046758493, 1504614657169716311674, 48848750173492332588525

Offset: 1

Author

Christian G. Bower, Nov 15 1998

Keywords

• nonn
• eigen

Comments

Nodes are numbered and the numbers increase as you move away from the root to any point on the same polygon.

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301 and Chapter 5.
• F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• F. Harary and R. Z. Norman, The Dissimilarity Characteristic of Husimi Trees, Annals of Mathematics, 58 1953, pp. 134-141.
• F. Harary and G. E. Uhlenbeck, On the Number of Husimi Trees, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953.
• Index entries for sequences related to cacti
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A035082-A035088.

Programs

Maple

A:= proc(n) option remember; if n<=1 then x else convert(series(Int(exp(A(n-1)^2/ (2-2*A(n-1))), x), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..22); # Alois P. Heinz, Aug 22 2008

Mathematica

max = 22; sy = Series[Integrate[E^(-(y^2/(2-2*y))), y], {y, 0, max}]; sx = Normal[ InverseSeries[sy, x]]; a[n_] := Coefficient[sx, x, n]*n!; Table[a[n], {n, 1, max }] (* Jean-François Alcover, Feb 24 2015 *)

Formula

E.g.f. satisfies A'(x) = exp(A(x)^2/(2-2*A(x))).