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

1,4

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

Formula

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

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 *)

Crossrefs

Cf. A035082-A035088.

Sequence in context: A332621 A091346 A305550 * A215852 A105797 A278189

Adjacent sequences: A035083 A035084 A035085 * A035087 A035088 A035089

Keyword

nonn,eigen

Author

Christian G. Bower, Nov 15 1998

Status

approved