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

%I A056066 #10 Sep 09 2013 19:14:39
%S A056066 0,1,3,28,570,22568,1682352,237014512,64144890960,33877404737792,
%T A056066 35289907832496768,72958473002707495168,300387071466709317941760,
%U A056066 2467720611903398552604259328,40493022471111759715270671578112,1327970521286614645847457853386207232
%N A056066 Expansion of log( dC(x)/dx ), C(x) = e.g.f. for labeled connected graphs (A001187).
%C A056066 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
%D A056066 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 16, Eq. (1.3.3).
%H A056066 Alois P. Heinz, <a href="/A056066/b056066.txt">Table of n, a(n) for n = 0..80</a>
%F A056066 E.g.f.: A(2x) - A(x) where A(x) is the e.g.f. for A001187. - _Geoffrey Critzer_, Sep 09 2013
%p A056066 b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
%p A056066       add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
%p A056066     end:
%p A056066 a:= proc(n) option remember; `if`(n=0, 0, b(n+1)-
%p A056066       add(k*binomial(n, k)*b(n+1-k)*a(k), k=1..n-1)/n)
%p A056066     end:
%p A056066 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 09 2013
%t A056066 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 *)
%K A056066 nonn
%O A056066 0,3
%A A056066 _N. J. A. Sloane_, Jul 29 2000