A327370 Number of labeled simple graphs with n vertices and exactly n - 1 endpoints (vertices of degree 1).

0, 1, 0, 6, 4, 50, 66, 532, 1016, 6876, 16750, 104456, 303612, 1821976, 6067166, 35857200, 133160176, 785514512, 3192117966, 18948962656, 83099447300, 498931946016, 2336474411062, 14234346694976, 70598633745576, 437304764440000, 2282139344678726, 14390600621415552

Offset: 0

Author

Gus Wiseman, Sep 04 2019

Keywords

• nonn

Comments

Graphs consist of zero or more paths on two nodes each and either a single isolated node or a star with two or more peripheral nodes. - Andrew Howroyd, Sep 05 2019

Examples

			The a(4) = 4 edge-sets:
  {12,13,14}
  {12,23,24}
  {13,23,34}
  {14,24,34}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Column k = n - 1 of A327369.

The unlabeled version is A028242.

Cf. A006125, A059167, A100743, A245797, A294217, A327227, A327377.

Programs

Maple

f:= gfun:-rectoproc({(n-1)*(n-2)*a(n)-n*(n-3)*(n-2)*a(n-1)-n*(n-1)^2*a(n-2)+(2*n-7)*n*(n-1)*(n-2)*a(n-3)-n*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0, a(0)=0, a(1)=1, a(2)=0, a(3)=6, a(4)=4},a(n),remember):
map(f, [$0..40]); # Robert Israel, Sep 06 2019

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==n-1&]],{n,0,5}]
With[{nn=30},CoefficientList[Series[x Exp[x^2/2](Exp[x]-x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 28 2022 *)

PARI

seq(n)={Vec(serlaplace(x*exp(x^2/2 + O(x^n))*(exp(x + O(x^n))-x)), -(n+1))} \\ Andrew Howroyd, Sep 05 2019

Formula

E.g.f.: x*exp(x^2/2)*(exp(x) - x). - Andrew Howroyd, Sep 05 2019

(n-1)*(n-2)*a(n) - n*(n-3)*(n-2)*a(n-1) - n*(n-1)^2*a(n-2) + (2*n-7)*n*(n-1)*(n-2)*a(n-3) - n*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) = 0. - Robert Israel, Sep 06 2019

Extensions

Terms a(8) and beyond from Andrew Howroyd, Sep 05 2019