OFFSET
0,3
COMMENTS
a(n) is the number of ways to designate a node in each connected component over all simple labeled graphs on n nodes.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..80
FORMULA
E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A053549.
MAPLE
g:= 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)*g(k), k=1..n-1)/n)
end:
a:= proc(n) option remember; `if`(n=0, 1, add(
binomial(n-1, j-1) *j*g(j) *a(n-j), j=1..n))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Mar 17 2015
MATHEMATICA
nn=20; a=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[Exp[x D[Log[a], x]], {x, 0, nn}], x]
PROG
(PARI) seq(n)={Vec(serlaplace(exp(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))))))} \\ Andrew Howroyd, Jun 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Oct 20 2011
EXTENSIONS
a(6), a(10) corrected by Alois P. Heinz, Mar 18 2015
STATUS
approved