A245797 The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1.

0, 1, 6, 49, 710, 19011, 954184, 90154415, 16108626420, 5481798833245, 3582369649269620, 4532127781040045649, 11177949079089720090800, 54050029251399545975868271, 514598463471970554205910304780, 9677402372862708729859372687791391

Offset: 1

Author

Chai Wah Wu, Aug 01 2014

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50

Crossrefs

Equal to row sums of A245796.

The covering case is A327227.

The connected case is A327362.

The generalization to set-systems is A327228.

BII-numbers of set-systems with minimum degree 1 are A327105.

Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327229, A327230.

Programs

Mathematica

m = 16;
egf = Exp[x^2/2]*Sum[2^Binomial[n, 2]*(x/Exp[x])^n/n!, {n, 0, m}];
A059167[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;
a[n_] := 2^(n(n-1)/2) - A059167[n];
Array[a, m] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}] (* Gus Wiseman, Sep 11 2019 *)

Formula

a(n) = 2^(n*(n+1)/2) - A059167(n).

Binomial transform of A327227 (assuming a(0) = 0).

Extensions

a(9)-a(16) from Andrew Howroyd, Oct 26 2017