%I #33 Aug 25 2024 20:55:53

%S 1,2,12,112,1450,23976,482944,11472896,314061948,9734500000,

%T 336998573296,12888244482048,539640296743288,24552709165722752,

%U 1206192446775000000,63633506348182798336,3587991568046845781776,215334327830586721473024,13705101790650454900938688

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

%C Also labeled involution rooted trees.

%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>

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

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

%F a(n) = i^(n-1)*n^((n-1)/2)*He_{n-1}(-sqrt(-n)), i=sqrt(-1), He_k unitary Hermite polynomial (cf. A066325).

%F a(n) = Sum_{k = ceiling((n-1)/2)...n-1} (n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1). - _Vladimir Kruchinin_, Aug 07 2012

%F a(n) ~ 2^(n+1/2) * n^(n-1) * exp((sqrt(5)-3)*n/4) / (sqrt(5+sqrt(5)) * (sqrt(5)-1)^n). - _Vaclav Kotesovec_, Jan 08 2014

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

%o (Maxima) a(n):=sum(((n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1)),k,ceiling((n-1)/2),n-1); /* _Vladimir Kruchinin_, Aug 07 2012 */

%o (PARI) x='x+O('x^66);

%o Vec(serlaplace(serreverse(x/exp(x^2/2+x)))) /* _Joerg Arndt_, Jan 25 2013 */

%Y a(n) = A091485(n)*n. Cf. A032035, A066325, A091486.

%K nonn,eigen

%O 1,2

%A _Christian G. Bower_, Jan 13 2004