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A221864 Number of forests (sets) of rooted labeled trees on {1,2,...,n} such that the node with label 1 is in the same rooted tree as the node with label 2.

%I A221864 #25 Aug 31 2014 04:55:11
%S A221864 0,0,2,11,88,930,12254,193736,3576564,75552560,1797906490,47601571968,
%T A221864 1388102588048,44210926113536,1527152437488150,56867807937459200,
%U A221864 2271048787266451756,96826981390532388864,4389830567318703987314,210886652765343862784000
%N A221864 Number of forests (sets) of rooted labeled trees on {1,2,...,n} such that the node with label 1 is in the same rooted tree as the node with label 2.
%H A221864 Alois P. Heinz, <a href="/A221864/b221864.txt">Table of n, a(n) for n = 0..200</a>
%F A221864 a(n) = Sum_{k=0..n-2} C(n-2,k)*(k+2)^(k+1)*(n-k-1)^(n-k-3).
%F A221864 E.g.f.: Double integral of T''(x)*exp(T(x)) dx^2 where T(x) is the e.g.f. for A000169.
%F A221864 a(n) ~ exp(1) * n^(n-1) * (1 - sqrt(Pi/(2*n))). - _Vaclav Kotesovec_, Aug 31 2014
%e A221864 a(3) = 9 + 2 = 11 because we have A000169(3) = 9 forests composed of a single rooted tree and 2 forests composed of two rooted trees:
%e A221864 1'-2  3,  2'-1  3 where the root is indicated with '.
%p A221864 a:= n-> add(binomial(n-2, k)*(k+2)^(k+1)*(n-k-1)^(n-k-3), k=0..n-2):
%p A221864 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 11 2013
%t A221864 nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Integrate[Integrate[D[D[t,x],x]Exp[t],x],x],{x,0,nn}],x]
%t A221864 Flatten[{0,0,CoefficientList[Series[-(2 + LambertW[-x]) / (x^3*(1 + 1/LambertW[-x])^3), {x, 0, 20}], x] * Range[0, 20]!}] (* _Vaclav Kotesovec_, Aug 31 2014 *)
%Y A221864 Cf. A000169, A000272.
%K A221864 nonn
%O A221864 0,3
%A A221864 _Geoffrey Critzer_, Apr 10 2013