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 A327227 Number of labeled simple graphs covering n vertices with at least one endpoint/leaf. 22
 0, 0, 1, 3, 31, 515, 15381, 834491, 83016613, 15330074139, 5324658838645, 3522941267488973, 4489497643961740521, 11119309286377621015089, 53893949089393110881259181, 513788884660608277842596504415, 9669175277199248753133328740702449 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Covering means there are no isolated vertices. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge. Also graphs with minimum vertex-degree 1. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 FORMULA Inverse binomial transform of A245797, if we assume A245797(0) = 0. EXAMPLE The a(4) = 31 edge-sets:   {12,34}  {12,13,14}  {12,13,14,23}   {13,24}  {12,13,24}  {12,13,14,24}   {14,23}  {12,13,34}  {12,13,14,34}            {12,14,23}  {12,13,23,24}            {12,14,34}  {12,13,23,34}            {12,23,24}  {12,14,23,24}            {12,23,34}  {12,14,24,34}            {12,24,34}  {12,23,24,34}            {13,14,23}  {13,14,23,34}            {13,14,24}  {13,14,24,34}            {13,23,24}  {13,23,24,34}            {13,23,34}  {14,23,24,34}            {13,24,34}            {14,23,24}            {14,23,34}            {14,24,34} MATHEMATICA Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}] CROSSREFS Column k=1 of A327366. The non-covering version is A245797. The unlabeled version is A324693. The generalization to set-systems is A327229. BII-numbers of set-systems with minimum degree 1 are A327105. Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230. Sequence in context: A212917 A223993 A143637 * A245109 A121563 A261471 Adjacent sequences:  A327224 A327225 A327226 * A327228 A327229 A327230 KEYWORD nonn AUTHOR Gus Wiseman, Sep 01 2019 STATUS approved

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Last modified January 27 08:06 EST 2021. Contains 340460 sequences. (Running on oeis4.)