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A052512
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Number of rooted labeled trees of height at most 2.
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10
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0, 1, 2, 9, 40, 205, 1176, 7399, 50576, 372537, 2936080, 24617131, 218521128, 2045278261, 20112821288, 207162957135, 2228888801056, 24989309310961, 291322555295904, 3524580202643155, 44176839081266360, 572725044269255661, 7668896804574138232, 105920137923940473079
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OFFSET
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0,3
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COMMENTS
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Equivalently, number of mappings f from a set of n elements into itself such that f o f (f applied twice) is constant. - Robert FERREOL, Mar 05 2016
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LINKS
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FORMULA
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E.g.f.: x*exp(x*exp(x)).
a(n) = Sum_{k=0..n-1} n*C(n-1,k)*(n-k-1)^k. - Alois P. Heinz, Mar 15 2013
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EXAMPLE
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For n = 3 the a(3) = 9 mappings from {a,b,c} into itself are:
f_1(a) = f_1(b) = f_1(c) = a
f_2(c) = b, f_2(b) = f_2(a) = a
f_3(b) = c, f_3(c) = f_3(a) = a
and 6 others, associated to b and c.
(End)
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MAPLE
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spec := [S, {S=Prod(Z, Set(T1)), T2=Z, T1=Prod(Z, Set(T2))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
a:= n-> n*add(binomial(n-1, k)*(n-k-1)^k, k=0..n-1);
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MATHEMATICA
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nn=20; a=x Exp[x]; Range[0, nn]! CoefficientList[Series[x Exp[a], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 19 2012 *)
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PROG
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(PARI)
N=33; x='x+O('x^N);
egf=x*exp(x*exp(x));
v=Vec(serlaplace(egf));
vector(#v+1, n, if(n==1, 0, v[n-1]))
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x*Exp(x*Exp(x)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019
(Sage) m = 20; T = taylor(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
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CROSSREFS
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Cf. A000248 (forests with n nodes and height at most 1).
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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