A052322 Number of labeled rooted trees with a forbidden limb of length 3.

1, 2, 9, 40, 385, 4536, 66409, 1127792, 21981537, 483858640, 11873508361, 321497975448, 9522483900241, 306292854886760, 10632656242583145, 396223803663328096, 15776491521834720961, 668460175137505993248, 30030668624358362706697, 1425868954034374729854920

Offset: 1

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn
• eigen

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..390
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A002955, A002988-A002992, A052319-A052329.

Programs

Mathematica

Rest[CoefficientList[Series[-LambertW[-x*E^(-x^3)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

Formula

E.g.f. satisfies A(x) = x*exp(A(x) - x^3).

E.g.f: -LambertW(-x*exp(-x^3)). - Vaclav Kotesovec, Jan 08 2014

a(n) ~ sqrt(1+LambertW(-3*exp(-3))) * n^(n-1) * exp(n/3*LambertW(-3*exp(-3))). - Vaclav Kotesovec, Jan 08 2014