A275488 Number of labeled forests of (free) trees such that exactly one tree is a path.

1, 1, 3, 12, 80, 810, 10857, 174944, 3243060, 67859010, 1586109305, 41085509652, 1170954002946, 36469499267474, 1233416773419495, 45037748851872240, 1766375778253548392, 74067278799492363330, 3306928891056821667045, 156635771633727023132300

Offset: 1

Author

Geoffrey Critzer, Jul 30 2016

Keywords

• nonn

Comments

We could call such a graph a path through a forest.

Examples

			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.
		

References

• J. Harris, J. Hirst, M. Mossinghoff, Combinatorics and Graph Theory, Springer, 2010, page 34.

Crossrefs

Cf. A001858, A011800.

Programs

Mathematica

nn = 20; b[z_] := 1/((1 - z) 2) - 1/2 + z/2;
t[z_] := z + Sum[n^(n - 2) z^n/n!, {n, 2, nn}];
Drop[Range[0, nn]! CoefficientList[Series[b[z] Exp[t[z] - b[z]], {z, 0, nn}], z], 1]

Formula

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.

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