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A091481 Number of labeled rooted 2,3 cacti (triangular cacti with bridges). 2
1, 2, 12, 112, 1450, 23976, 482944, 11472896, 314061948, 9734500000, 336998573296, 12888244482048, 539640296743288, 24552709165722752, 1206192446775000000, 63633506348182798336, 3587991568046845781776, 215334327830586721473024, 13705101790650454900938688 (list; graph; refs; listen; history; text; internal format)
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

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)