A052320 Number of labeled trimmed trees with n nodes.

1, 1, 1, 0, 4, 5, 96, 1057, 14848, 235881, 4234240, 84815621, 1877090304, 45524670061, 1201345331200, 34283233751145, 1052350187831296, 34582597023733073, 1211614017182760960, 45088964565749965837

Offset: 0

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn

Comments

A trimmed tree is a tree with a forbidden limb of length 2.

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for sequences related to trees

Crossrefs

Cf. A002955, A002988-A002992, A052318-A052329.

Programs

Mathematica

CoefficientList[Series[1+x^3/2+x^4/2-LambertW[-x/E^(x^2)]-(LambertW[-x/E^(x^2)])^2/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

Formula

E.g.f.: 1 + x^3/2 + x^4/2 + B(x) - B(x)^2/2 where B(x) is e.g.f. of A052318.

a(n) ~ (1+LambertW(-2*exp(-2)))^(3/2) * exp((n*LambertW(-2*exp(-2)))/2) * n^(n-2). - Vaclav Kotesovec, Mar 29 2014