A038053 Number of labeled planted trees with 2-colored leaves.

0, 4, 12, 96, 1120, 17280, 330624, 7540736, 199544832, 6006988800, 202646118400, 7570772656128, 310240496517120, 13834761553313792, 666909048381112320, 34555424387503226880, 1915099718255940468736

Offset: 1

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for sequences related to rooted trees

Programs

Mathematica

CoefficientList[Series[(2+LambertW[-x*E^x])*(x-LambertW[-x*E^x])/(1+ LambertW[-x*E^x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

PARI

x='x+O('x^30); concat([0], Vec(serlaplace( (2+lambertw(-x*exp(x))) *(x-lambertw(-x*exp(x)))/(1+lambertw(-x*exp(x))) ))) \\ G. C. Greubel, Sep 09 2018

Formula

A038049 shifted right and multiplied by n.

E.g.f. (for offset 0): (2+B(x))*(x-B(x))/(1+B(x)) where B(x) = LambertW(-x*exp(x)). - Vladeta Jovovic, Mar 08 2003

a(n) ~ sqrt(LambertW(exp(-1))+1) * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^(n-1)). - Vaclav Kotesovec, Mar 29 2014