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%I #23 Sep 08 2022 08:44:59
%S 0,1,2,9,64,505,4536,46249,526352,6604497,90466480,1341571561,
%T 21392282088,364715915161,6616327512536,127187163197865,
%U 2581443127409056,55143025567270561,1236226458392407008,29012548251081127753,711157579030313374520,18169564436494014726441
%N Number of labeled trees of height at most 3.
%H G. C. Greubel, <a href="/A052513/b052513.txt">Table of n, a(n) for n = 0..400</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=59">Encyclopedia of Combinatorial Structures 59</a>
%F E.g.f.: x*exp(x*exp(x*exp(x))).
%p 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);
%t 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 *)
%o (PARI)
%o N=33; x='x+O('x^N);
%o egf=x*exp(x*exp(x*exp(x)));
%o v=Vec(serlaplace(egf));
%o vector(#v+1,n,if(n==1,0,v[n-1]))
%o /* _Joerg Arndt_, Sep 15 2012 */
%o (Magma) m:=20; R<x>:=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
%o (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
%Y Cf. A000552.
%Y Cf. A052514 (height at most 4).
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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