A091481 Number of labeled rooted 2,3 cacti (triangular cacti with bridges).

1, 2, 12, 112, 1450, 23976, 482944, 11472896, 314061948, 9734500000, 336998573296, 12888244482048, 539640296743288, 24552709165722752, 1206192446775000000, 63633506348182798336, 3587991568046845781776, 215334327830586721473024, 13705101790650454900938688

Offset: 1

Author

Christian G. Bower, Jan 13 2004

Keywords

• nonn
• eigen

Comments

Also labeled involution rooted trees.

References

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

Links

• Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
• Index entries for sequences related to cacti
• Index entries for sequences related to rooted trees

Crossrefs

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

Programs

Mathematica

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

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 */

PARI

x='x+O('x^66);
Vec(serlaplace(serreverse(x/exp(x^2/2+x)))) /* Joerg Arndt, Jan 25 2013 */

Formula

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

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

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

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