A035351 - OEIS

%I #28 Dec 19 2015 10:54:41

%S 1,2,12,124,1810,34056,783874,21331136,669902076,23845794400,

%T 948733833256,41721533664768,2009539243299328,105209055401980544,

%U 5948937678563109000,361296961279074942976,23456120142707873968336,1621073894248128387746304

%N Number of labeled rooted polygonal cacti with bridges (mixed Husimi trees) with n nodes.

%H Alois P. Heinz, <a href="/A035351/b035351.txt">Table of n, a(n) for n = 1..100</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>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

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

%F a(n) ~ (1-s)^2 * sqrt(2/((2-s)*(1+s-s^2))) * n^(n-1) / (s * exp((2+(s-4)*s)/(2-2*s)))^n, where s = 1/3*(4-2/(-17+3*sqrt(33))^(1/3) + (-17+3*sqrt(33))^(1/3)) = 0.456310987307923638429144... is the root of the equation 6*s - 4*s^2 + s^3 = 2. - _Vaclav Kotesovec_, Jan 08 2014

%p A:= proc(n) option remember; if n<=0 then x else x* exp((2*A(n-1) -A(n-1)^2)/ (2-2*A(n-1))) fi end: a:= n-> coeff(series(A(n-1), x=0,n+1), x,n)*n!: seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 20 2008

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

%Y Cf. A000083, A000237, A000314, A035082, A035349-A035357.

%K nonn,eigen

%O 1,2

%A _Christian G. Bower_, Nov 15 1998

%E More terms from _Alois P. Heinz_, Aug 20 2008