A038050 Number of labeled rooted trees with 3-colored leaves.

3, 6, 45, 504, 7785, 153468, 3681909, 104126256, 3392064945, 125089571700, 5151335388309, 234322765501608, 11668410187187481, 631335472193760012, 36881146426978035765, 2313552152470193124192, 155107536736245864549345

Offset: 1

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn
• eigen

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)

Links

• Index entries for sequences related to rooted trees
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000169, A029857.

Cf. A038049.

Programs

Mathematica

Rest[CoefficientList[Series[2*x-LambertW[-x*E^(2*x)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)

Formula

Divides by n and shifts left under exponential transform.

E.g.f.: 2*x - LambertW(-x*exp(2*x)). - Vladeta Jovovic, Mar 09 2003

a(n) = Sum_{k=0..n} (binomial(n, k)*2^k*(n-k)^(n-1)).

a(n) ~ sqrt(1+LambertW(2*exp(-1))) * (2*exp(-1)/LambertW(2*exp(-1)))^n * n^(n-1). - Vaclav Kotesovec, Oct 05 2013