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A056066 Expansion of log( dC(x)/dx ), C(x) = e.g.f. for labeled connected graphs (A001187). 1
0, 1, 3, 28, 570, 22568, 1682352, 237014512, 64144890960, 33877404737792, 35289907832496768, 72958473002707495168, 300387071466709317941760, 2467720611903398552604259328, 40493022471111759715270671578112, 1327970521286614645847457853386207232 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

FORMULA

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

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 *)

CROSSREFS

Sequence in context: A156315 A248571 A062497 * A174483 A092985 A181588

Adjacent sequences:  A056063 A056064 A056065 * A056067 A056068 A056069

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 29 2000

STATUS

approved

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Last modified May 26 17:17 EDT 2019. Contains 323597 sequences. (Running on oeis4.)