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

1,1

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

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

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 *)

Crossrefs

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

Sequence in context: A378581 A141307 A190655 * A151817 A265937 A038058

Adjacent sequences: A038046 A038047 A038048 * A038050 A038051 A038052

Keyword

nonn,eigen

Author

Christian G. Bower, Jan 04 1999

Status

approved