A035087 Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.

1, 0, 3, 12, 135, 1440, 20895, 342720, 6585705, 142430400, 3449279295, 92207808000, 2699909867655, 85900402748160, 2951318065570875, 108894519775641600, 4294542443185019025, 180277244225580902400, 8025792422657714379675, 377695544010698833920000

Offset

1,3

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.

Harary and E. M. Palmer, Graphical Enumeration, p. 71

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..400

Index entries for sequences related to cacti

Index entries for sequences related to rooted trees

Formula

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

a(n) ~ (1-s)^2 * sqrt(2/(6-11*s+4*s^2)) * n^(n-1) / (s * exp(1 - s^2/(2*(1-s))))^n, where s = 0.5391888728108891165... is the root of the equation 2-4*s+s^3=0. - Vaclav Kotesovec, Jan 08 2014

Maple

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)^2/ (2-2*A(n-1))), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..30); # Alois P. Heinz, Aug 22 2008

Mathematica

Rest[CoefficientList[InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

Crossrefs

Cf. A035082-A035088.

Sequence in context: A152544 A280115 A264149 * A056426 A056417 A363412

Adjacent sequences: A035084 A035085 A035086 * A035088 A035089 A035090

Keyword

nonn,eigen

Author

Christian G. Bower, Nov 15 1998

Status

approved