A035086 - OEIS

%I #23 Feb 24 2015 10:27:50

%S 1,0,1,3,19,135,1204,12537,150556,2043930,30969211,517973148,

%T 9478800604,188381470095,4040440921699,93020386382742,

%U 2287969523647171,59877222907995675,1661259526266784171,48705364034046758493,1504614657169716311674,48848750173492332588525

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

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

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301 and Chapter 5.

%D F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

%H Alois P. Heinz, <a href="/A035086/b035086.txt">Table of n, a(n) for n = 1..200</a>

%H F. Harary and R. Z. Norman, <a href="http://www.jstor.org/stable/1969824">The Dissimilarity Characteristic of Husimi Trees</a>, Annals of Mathematics, 58 1953, pp. 134-141.

%H F. Harary and G. E. Uhlenbeck, <a href="http://www.pnas.org/content/39/4/315.full.pdf">On the Number of Husimi Trees</a>, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953.

%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) = exp(A(x)^2/(2-2*A(x))).

%p 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

%t 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 *)

%Y Cf. A035082-A035088.

%K nonn,eigen

%O 1,4

%A _Christian G. Bower_, Nov 15 1998