A371633 Number of ways to choose a simple labeled graph on [n], then partition the vertex set into independent sets, then choose a vertex from each independent set.

1, 1, 4, 35, 740, 34629, 3581894, 802937479, 386655984648, 396751196145673, 862046936883049482, 3946154005780155709451, 37896676657907955726032908, 760791471852690599411320471565, 31830237745009483676211065390546958, 2768049771339996987073597682850993569807

Offset: 0

Author

Geoffrey Critzer, Jun 06 2024

Keywords

• nonn

Comments

An independent set is a set of vertices in a graph, no two of which are adjacent.

Links

• R. P. Stanley, Exponential structures

Crossrefs

Cf. A000248, A240936, A058843, A011266.

Programs

Mathematica

nn = 14; B[n_] := n! 2^Binomial[n, 2];ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /.Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}];Table[B[n], {n, 0, nn}] CoefficientList[Series[Exp[ggf[x Exp[x]]], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n)*x^n/A011266(n) = exp(f(x)) where f(x) = Sum_{n>=1} n*x^n/A011266(n).