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 A245797 The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1. 22

%I

%S 0,1,6,49,710,19011,954184,90154415,16108626420,5481798833245,

%T 3582369649269620,4532127781040045649,11177949079089720090800,

%U 54050029251399545975868271,514598463471970554205910304780,9677402372862708729859372687791391

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

%H Andrew Howroyd, <a href="/A245797/b245797.txt">Table of n, a(n) for n = 1..50</a>

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

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

%t m = 16;

%t egf = Exp[x^2/2]*Sum[2^Binomial[n, 2]*(x/Exp[x])^n/n!, {n, 0, m}];

%t A059167[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;

%t a[n_] := 2^(n(n-1)/2) - A059167[n];

%t Array[a, m] (* _Jean-François Alcover_, Feb 23 2019 *)

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}] (* _Gus Wiseman_, Sep 11 2019 *)

%Y Equal to row sums of A245796.

%Y The covering case is A327227.

%Y The connected case is A327362.

%Y The generalization to set-systems is A327228.

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

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

%K nonn

%O 1,3

%A _Chai Wah Wu_, Aug 01 2014

%E a(9)-a(16) from _Andrew Howroyd_, Oct 26 2017

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Last modified April 10 20:35 EDT 2021. Contains 342856 sequences. (Running on oeis4.)