A052317 Number of labeled trees with n nodes and 2-colored internal (non-leaf) nodes.

1, 1, 1, 6, 56, 730, 12372, 259574, 6511920, 190413234, 6364960940, 239556803662, 10028763883272, 462366507311306, 23282257730716740, 1271520006077859750, 74865320814990626912, 4727699146425478764898, 318763676354643090937692, 22856568223852002933212798

Offset: 0

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn

Links

• Index entries for sequences related to trees

Crossrefs

Cf. A004113, A004114, A052316.

Programs

Mathematica

CoefficientList[Series[1+(1-x)*(-x-LambertW[-2*x*E^(-x)])-(-x-LambertW[-2*x*E^(-x)])^2/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

Formula

E.g.f.: 1 + B(x) - x*B(x) - B(x)^2/2 where B(x) is e.g.f. of A052316.

a(n) ~ (1+LambertW(-exp(-1)/2))^(3/2) * n^(n-2) / (exp(n)*(-LambertW(-exp(-1)/2))^n). - Vaclav Kotesovec, Oct 05 2013