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A275488 Number of labeled forests of (free) trees such that exactly one tree is a path. 0
1, 1, 3, 12, 80, 810, 10857, 174944, 3243060, 67859010, 1586109305, 41085509652, 1170954002946, 36469499267474, 1233416773419495, 45037748851872240, 1766375778253548392, 74067278799492363330, 3306928891056821667045, 156635771633727023132300 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..20.

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

Cf. A001858, A011800.

Sequence in context: A023879 A084565 A323634 * A304561 A303227 A182166

Adjacent sequences:  A275485 A275486 A275487 * A275489 A275490 A275491

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jul 30 2016

STATUS

approved

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Last modified September 20 10:44 EDT 2019. Contains 327229 sequences. (Running on oeis4.)