%I #19 Feb 27 2016 15:57:37
%S 1,0,3,12,135,1440,20895,342720,6585705,142430400,3449279295,
%T 92207808000,2699909867655,85900402748160,2951318065570875,
%U 108894519775641600,4294542443185019025,180277244225580902400,8025792422657714379675,377695544010698833920000
%N Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
%D Harary and E. M. Palmer, Graphical Enumeration, p. 71
%D F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141
%D F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953
%H Alois P. Heinz, <a href="/A035087/b035087.txt">Table of n, a(n) for n = 1..400</a>
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f. satisfies A(x)=x*exp(A(x)^2/(2-2*A(x))).
%F 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
%p 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
%t Rest[CoefficientList[InverseSeries[Series[E^(x^2/(2*(x-1)))*x,{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 08 2014 *)
%Y Cf. A035082-A035088.
%K nonn,eigen
%O 1,3
%A _Christian G. Bower_, Nov 15 1998