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

1,2

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

Table of n, a(n) for n=1..19.

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

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

Mathematica

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

Prog

(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 */

Crossrefs

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

Sequence in context: A227460 A316651 A330654 * A053312 A091854 A141141

Adjacent sequences: A091478 A091479 A091480 * A091482 A091483 A091484

Keyword

nonn,eigen

Author

Christian G. Bower, Jan 13 2004

Status

approved