cp's OEIS Frontend

A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

%I A100743 #20 Sep 04 2019 21:44:27
%S A100743 1,0,0,1,10,253,12068,1052793,169505868,51046350021,29184353055900,
%T A100743 32122563615242615,68867440268165982320,290155715157676330952559,
%U A100743 2417761590648159731258579164,40013923822242935823157820555477,1318910080336893719646370269435043184
%N A100743 Number of labeled n-vertex graphs without vertices of degree <=1.
%H A100743 Andrew Howroyd, <a href="/A100743/b100743.txt">Table of n, a(n) for n = 0..50</a>
%F A100743 E.g.f.: exp(-x+x^2/2)*(Sum_{n>=0} 2^(n*(n-1)/2)*(x/exp(x))^n/n!). - _Vladeta Jovovic_, Jan 26 2006
%F A100743 Exponential transform of A059166. - _Gus Wiseman_, Aug 18 2019
%F A100743 Inverse binomial transform of A059167. - _Gus Wiseman_, Sep 02 2019
%e A100743 From _Gus Wiseman_, Aug 18 2019: (Start)
%e A100743 The a(4) = 10 edge-sets:
%e A100743   {12,13,24,34}
%e A100743   {12,14,23,34}
%e A100743   {13,14,23,24}
%e A100743   {12,13,14,23,24}
%e A100743   {12,13,14,23,34}
%e A100743   {12,13,14,24,34}
%e A100743   {12,13,23,24,34}
%e A100743   {12,14,23,24,34}
%e A100743   {13,14,23,24,34}
%e A100743   {12,13,14,23,24,34}
%e A100743 (End)
%t A100743 m = 13;
%t A100743 egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}];
%t A100743 s = egf + O[x]^(m+1);
%t A100743 a[n_] := n!*SeriesCoefficient[s, n];
%t A100743 Table[a[n], {n, 0, m}] (* _Jean-François Alcover_, Feb 23 2019 *)
%t A100743 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}] (* _Gus Wiseman_, Aug 18 2019 *)
%o A100743 (PARI) seq(n)={Vec(serlaplace(exp(-x + x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1)/2)*(x/exp(x + O(x^n)))^k/k!)))} \\ _Andrew Howroyd_, Sep 04 2019
%Y A100743 Graphs without isolated nodes are A006129.
%Y A100743 The connected case is A059166.
%Y A100743 Graphs without endpoints are A059167.
%Y A100743 Labeled graphs with endpoints are A245797.
%Y A100743 The unlabeled version is A261919.
%Y A100743 Cf. A095983, A136284, A322395, A327079, A327107, A327227, A327229, A327230.
%Y A100743 Cf. A095983, A322395.
%K A100743 nonn
%O A100743 0,5
%A A100743 Goran Kilibarda, Zoran Maksimovic, _Vladeta Jovovic_, Jan 03 2005
%E A100743 Terms a(14) and beyond from _Andrew Howroyd_, Sep 04 2019