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

Author

Christian G. Bower, Nov 15 1998

Keywords

• nonn
• eigen

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

Crossrefs

Cf. A035082-A035088.

Programs

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 *)

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