A038049 Number of labeled rooted trees with 2-colored leaves.

2, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320

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

• Alois P. Heinz, Table of n, a(n) for n = 1..150
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000169, A029856, A038050, A038054, A088789.

Programs

Maple

a:= n-> add(binomial(n, k)*(n-k)^(n-1), k=0..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 30 2012

Mathematica

Table[n!*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,20}] (* Vaclav Kotesovec, Nov 30 2012 *)

Formula

Divides by n and shifts left under exponential transform.

E.g.f.: A(x) = x-LambertW(-x*exp(x)). - Vladeta Jovovic, Mar 08 2003

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

A(x) = 2*compositional inverse of 2*x/(1+exp(2*x)). - Peter Bala, Oct 14 2011

a(n) ~ n^(n-1) * sqrt((1+LambertW(1/e))) / (e*LambertW(1/e))^n. - Vaclav Kotesovec, Nov 30 2012