cp's OEIS Frontend

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

%I A327370 #19 Apr 28 2022 19:23:05
%S A327370 0,1,0,6,4,50,66,532,1016,6876,16750,104456,303612,1821976,6067166,
%T A327370 35857200,133160176,785514512,3192117966,18948962656,83099447300,
%U A327370 498931946016,2336474411062,14234346694976,70598633745576,437304764440000,2282139344678726,14390600621415552
%N A327370 Number of labeled simple graphs with n vertices and exactly n - 1 endpoints (vertices of degree 1).
%C A327370 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
%H A327370 Andrew Howroyd, <a href="/A327370/b327370.txt">Table of n, a(n) for n = 0..500</a>
%F A327370 E.g.f.: x*exp(x^2/2)*(exp(x) - x). - _Andrew Howroyd_, Sep 05 2019
%F A327370 (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
%e A327370 The a(4) = 4 edge-sets:
%e A327370   {12,13,14}
%e A327370   {12,23,24}
%e A327370   {13,23,34}
%e A327370   {14,24,34}
%p A327370 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):
%p A327370 map(f, [$0..40]); # _Robert Israel_, Sep 06 2019
%t A327370 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==n-1&]],{n,0,5}]
%t A327370 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 *)
%o A327370 (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
%Y A327370 Column k = n - 1 of A327369.
%Y A327370 The unlabeled version is A028242.
%Y A327370 Cf. A006125, A059167, A100743, A245797, A294217, A327227, A327377.
%K A327370 nonn
%O A327370 0,4
%A A327370 _Gus Wiseman_, Sep 04 2019
%E A327370 Terms a(8) and beyond from _Andrew Howroyd_, Sep 05 2019