OFFSET
2,2
COMMENTS
Also the number of labeled simple graphs with n vertices whose edge-set is connected. - Gus Wiseman, Sep 11 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..50
Eric Weisstein's World of Mathematics, Triangular Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
FORMULA
a(n) = Sum_{i=2..n} binomial(n,i) * A001187(i). - Andrew Howroyd, Jun 07 2017
E.g.f.: exp(x)*(-x + log(Sum_{k>=0} 2^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Sep 11 2019
EXAMPLE
From Gus Wiseman, Sep 11 2019: (Start)
The a(4) = 60 edge-sets:
{12} {12,13} {12,13,14} {12,13,14,23} {12,13,14,23,24}
{13} {12,14} {12,13,23} {12,13,14,24} {12,13,14,23,34}
{14} {12,23} {12,13,24} {12,13,14,34} {12,13,14,24,34}
{23} {12,24} {12,13,34} {12,13,23,24} {12,13,23,24,34}
{24} {13,14} {12,14,23} {12,13,23,34} {12,14,23,24,34}
{34} {13,23} {12,14,24} {12,13,24,34} {13,14,23,24,34}
{13,34} {12,14,34} {12,14,23,24}
{14,24} {12,23,24} {12,14,23,34}
{14,34} {12,23,34} {12,14,24,34}
{23,24} {12,24,34} {12,23,24,34}
{23,34} {13,14,23} {13,14,23,24}
{24,34} {13,14,24} {13,14,23,34}
{13,14,34} {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} {12,13,14,23,24,34}
{23,24,34}
(End)
MATHEMATICA
Table[With[{g = GraphData[{"Triangular", n}]}, Total[Boole[ConnectedGraphQ[Subgraph[g, #]] & /@ Subsets[VertexList[g]]]]], {n, 2, 5}] - 1
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k) * (n-k-1)/2)*g[k], {k, 1, n-1}]/n]; a[n_] := Sum[Binomial[n, i]*g[i], {i, 2, n}]; Table[a[n], {n, 2, 16}] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)
PROG
(PARI) seq(n)={Vec(serlaplace(exp(x + O(x*x^n))*(-x+log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Sep 11 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, May 29 2017
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Jun 07 2017
STATUS
approved