OFFSET
1,3
COMMENTS
We could call such a graph a path through a forest.
REFERENCES
J. Harris, J. Hirst, M. Mossinghoff, Combinatorics and Graph Theory, Springer, 2010, page 34.
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
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jul 30 2016
STATUS
approved
