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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 29 2000
STATUS
approved