A052514 - OEIS

%I #22 Sep 08 2022 08:44:59

%S 0,1,2,9,64,625,7056,89929,1284032,20351601,354648160,6736612201,

%T 138472331328,3061103815081,72391319923664,1823032999274985,

%U 48692068509655936,1374488205290880481,40877130077266074048

%N Number of labeled trees of height at most 4.

%H G. C. Greubel, <a href="/A052514/b052514.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=60">Encyclopedia of Combinatorial Structures 60</a>

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

%p spec := [S,{T2=Prod(Z,Set(T3)),S=Prod(Z,Set(T1)), T4=Z, T3=Prod(Z,Set(T4)), T1=Prod(Z,Set(T2))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t With[{nn=20},CoefficientList[Series[x*Exp[x*Exp[x*Exp[x*Exp[x]]]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 23 2018 *)

%o (PARI) my(x='x+O('x^20)); concat(0, Vec(serlaplace( x*exp(x*exp(x*exp(x*exp(x)))) ))) \\ _G. C. Greubel_, May 13 2019

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x*Exp(x*Exp(x*Exp(x*Exp(x)))) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, May 13 2019

%o (Sage) m = 20; T = taylor(x*exp(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

%Y Cf. A052513 (height at most 3).

%K easy,nonn

%O 0,3

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

%E Added "at most" in the title; by _Stanislav Sykora_, May 12 2012