A052323 Number of labeled trees with a forbidden limb of length 3.

1, 1, 1, 3, 4, 65, 576, 5887, 92464, 1680345, 34041520, 774906011, 19537590744, 540890740117, 16321259150392, 533305854910935, 18764822871806176, 707498057530634033, 28460428902580264416, 1216828054782241792435

Offset: 0

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn

Comments

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^4/2+x^5/2+x^6/2-LambertW[-x*E^(-x^3)]-(LambertW[-x*E^(-x^3)])^2/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

Formula

E.g.f.: 1+x^4/2+x^5/2+x^6/2+B(x)-B(x)^2/2 where B(x) is e.g.f. of A052322.

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