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A220690 Number of acyclic graphs on {1,2,...,n} such that the node with label 1 is in the same connected component (tree) as the node with label 2.

%I A220690 #14 Apr 13 2013 16:36:47
%S A220690 0,0,1,4,24,198,2110,27768,436656,8003950,167779068,3961727820,
%T A220690 104102329504,3013887239454,95338047836520,3272043459321328,
%U A220690 121106541865151040,4808924948167249302,203931444227955436816,9198925314402386788500,439809753701222702598528
%N A220690 Number of acyclic graphs on {1,2,...,n} such that the node with label 1 is in the same connected component (tree) as the node with label 2.
%H A220690 Alois P. Heinz, <a href="/A220690/b220690.txt">Table of n, a(n) for n = 0..200</a>
%F A220690 E.g.f.: Double integral of U''(x)*exp(U(x)) dx^2 where U(x) is the e.g.f. for A000272.
%F A220690 a(n) = Sum_{k=0..n-2} binomial(n-2,k)*(k+2)^k*A001858(n-k-2).
%p A220690 b:= proc(n) option remember; `if`(n=0, 1,
%p A220690       add(binomial(n-1, j) *(j+1)^(j-1) *b(n-1-j), j=0..n-1))
%p A220690     end:
%p A220690 a:= n-> add(binomial(n-2, k)*(k+2)^k*b(n-k-2), k=0..n-2):
%p A220690 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 13 2013
%t A220690 nn=20;u=Sum[n^(n-2)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Integrate[Integrate[D[D[u,x],x]Exp[u],x],x],{x,0,nn}],x]
%o A220690 (PARI)
%o A220690 N = 66;  x = 'x + O('x^N);
%o A220690 U = sum(n=1,N,n^(n-2)*x^n/n!);
%o A220690 egf = intformal(intformal( deriv(deriv(U)) * exp(U) ));
%o A220690 gf = serlaplace(egf) + 'c0;
%o A220690 v = Vec(gf);  v[1]-='c0;  v
%o A220690 /* _Joerg Arndt_, Apr 13 2013 */
%Y A220690 Cf. A221864.
%K A220690 nonn
%O A220690 0,4
%A A220690 _Geoffrey Critzer_, Apr 13 2013