A091485 - OEIS

%I #27 Aug 19 2023 19:04:07

%S 1,1,4,28,290,3996,68992,1434112,34895772,973450000,30636233936,

%T 1074020373504,41510792057176,1753764940408768,80412829785000000,

%U 3977094146761424896,211058327532167398928,11963018212810373415168,721321146876339731628352

%N Number of labeled 2,3 cacti (triangular cacti with bridges).

%C As _Alois P. Heinz_ has pointed out, the e.g.f in the Example section does not match the offset. However, the identity a(n) = A091481(n)/n holds with the present offset of 1. - _N. J. A. Sloane_, Jun 23 2017

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

%H Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.

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

%F a(n) = A091481(n)/n.

%F From _Paul D. Hanna_, Jun 01 2012: (Start)

%F E.g.f.: (1/x)*Series_Reversion( x/exp(x+x^2/2) ).

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

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

%F (End)

%F a(n+1) = n! * Sum_{k=0..n} (1/2)^(n-k) * (n+1)^(k-1) * binomial(k,n-k)/k!. - _Seiichi Manyama_, Aug 19 2023

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 290*x^4/4! + 3996*x^5/5! +...

%t CoefficientList[1/x InverseSeries[x/Exp[x+x^2/2]+O[x]^20], x] Range[0, 18]! (* _Jean-François Alcover_, Aug 06 2018 *)

%Y Cf. A091481, A091487, A000085.

%K nonn

%O 1,3

%A _Christian G. Bower_, Jan 14 2004