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%I #22 Aug 01 2016 08:34:31
%S 1,1,3,12,80,810,10857,174944,3243060,67859010,1586109305,41085509652,
%T 1170954002946,36469499267474,1233416773419495,45037748851872240,
%U 1766375778253548392,74067278799492363330,3306928891056821667045,156635771633727023132300
%N Number of labeled forests of (free) trees such that exactly one tree is a path.
%C We could call such a graph a path through a forest.
%D J. Harris, J. Hirst, M. Mossinghoff, Combinatorics and Graph Theory, Springer, 2010, page 34.
%F E.g.f.: B(x)*exp(T(x)-B(x)) where B(x) is the e.g.f. for A001710 - 1 and T(x) is the e.g.f. for A000272 - 1.
%F a(n) ~ (2*exp(1)-1) * exp((exp(-1)-exp(1)-1)/(2*(exp(1)-1))) * n^(n-2) / (2*(exp(1)-1)). - _Vaclav Kotesovec_, Jul 31 2016
%e a(1),a(2),a(3),a(4) are just a single path through an empty forest. a(5)=80 counts the 60 labelings of a path on 5 nodes and the 20 labelings of a path on 1 node and a star on 4 nodes.
%t nn = 20; b[z_] := 1/((1 - z) 2) - 1/2 + z/2;
%t t[z_] := z + Sum[n^(n - 2) z^n/n!, {n, 2, nn}];
%t Drop[Range[0, nn]! CoefficientList[Series[b[z] Exp[t[z] - b[z]], {z, 0, nn}], z], 1]
%Y Cf. A001858, A011800.
%K nonn
%O 1,3
%A _Geoffrey Critzer_, Jul 30 2016
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