A056066 Expansion of log( dC(x)/dx ), C(x) = e.g.f. for labeled connected graphs (A001187).

0, 1, 3, 28, 570, 22568, 1682352, 237014512, 64144890960, 33877404737792, 35289907832496768, 72958473002707495168, 300387071466709317941760, 2467720611903398552604259328, 40493022471111759715270671578112, 1327970521286614645847457853386207232

Offset: 0

Author

N. J. A. Sloane, Jul 29 2000

Keywords

• nonn

Comments

a(n) is the number of connected simple labeled graphs G on {1,2,...,n+1} such that G is still connected upon removal of the vertex n+1. Equivalently, a(n) is the number of ways to form a connected simple labeled graph on {1,2,...,n} and then select a nonempty subset of its vertices. This statement translates immediately via the symbolic method into the e.g.f. given below. - Geoffrey Critzer, Sep 09 2013

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 16, Eq. (1.3.3).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..80

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(n+1)-
      add(k*binomial(n, k)*b(n+1-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 09 2013

Mathematica

nn=14;f[x_]:=Log[Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}]]+1;Range[0,nn]!CoefficientList[Series[f[2x]-f[x],{x,0,nn}],x] (* Geoffrey Critzer, Sep 09 2013 *)

Formula

E.g.f.: A(2x) - A(x) where A(x) is the e.g.f. for A001187. - Geoffrey Critzer, Sep 09 2013