A052318 Number of labeled rooted trimmed trees with n nodes.

1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320

Offset: 1

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn
• eigen

Comments

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

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..400
• Index entries for sequences related to rooted trees

Crossrefs

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

Programs

Maple

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008

Mathematica

a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

Maxima

a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* Vladimir Kruchinin, Aug 07 2012 */

Formula

E.g.f. satisfies A(x) = x*exp(A(x) - x^2).

E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014

a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014