A052513 Number of labeled trees of height at most 3.

0, 1, 2, 9, 64, 505, 4536, 46249, 526352, 6604497, 90466480, 1341571561, 21392282088, 364715915161, 6616327512536, 127187163197865, 2581443127409056, 55143025567270561, 1236226458392407008, 29012548251081127753, 711157579030313374520, 18169564436494014726441

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 59

Crossrefs

Cf. A000552.

Cf. A052514 (height at most 4).

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x*Exp(x*Exp(x*Exp(x))) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019

Maple

spec := [S,{S=Prod(Z,Set(T1)), T2=Prod(Z,Set(T3)), T3=Z, T1=Prod(Z,Set(T2))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

nn=20;a=x Exp[x];b=x Exp[a];Range[0,nn]! CoefficientList[Series[x Exp[b],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 20 2012 *)

PARI

N=33;  x='x+O('x^N);
egf=x*exp(x*exp(x*exp(x)));
v=Vec(serlaplace(egf));
vector(#v+1,n,if(n==1,0,v[n-1]))
/* Joerg Arndt, Sep 15 2012 */

Sage

m = 20; T = taylor(x*exp(x*exp(x*exp(x))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: x*exp(x*exp(x*exp(x))).