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 A100743 Number of labeled n-vertex graphs without vertices of degree <=1. 13
 1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 FORMULA 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 Exponential transform of A059166. - Gus Wiseman, Aug 18 2019 Inverse binomial transform of A059167. - Gus Wiseman, Sep 02 2019 EXAMPLE From Gus Wiseman, Aug 18 2019: (Start) The a(4) = 10 edge-sets:   {12,13,24,34}   {12,14,23,34}   {13,14,23,24}   {12,13,14,23,24}   {12,13,14,23,34}   {12,13,14,24,34}   {12,13,23,24,34}   {12,14,23,24,34}   {13,14,23,24,34}   {12,13,14,23,24,34} (End) MATHEMATICA m = 13; egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}]; s = egf + O[x]^(m+1); a[n_] := n!*SeriesCoefficient[s, n]; Table[a[n], {n, 0, m}] (* Jean-François Alcover, Feb 23 2019 *) 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 *) PROG (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 CROSSREFS Graphs without isolated nodes are A006129. The connected case is A059166. Graphs without endpoints are A059167. Labeled graphs with endpoints are A245797. The unlabeled version is A261919. Cf. A095983, A136284, A322395, A327079, A327107, A327227, A327229, A327230. Cf. A095983, A322395. Sequence in context: A178689 A095983 A059166 * A251588 A126468 A336665 Adjacent sequences:  A100740 A100741 A100742 * A100744 A100745 A100746 KEYWORD nonn AUTHOR Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 03 2005 EXTENSIONS Terms a(14) and beyond from Andrew Howroyd, Sep 04 2019 STATUS approved

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Last modified May 9 02:35 EDT 2021. Contains 343685 sequences. (Running on oeis4.)