A198046 Exponential transform of A053549.

1, 1, 3, 19, 225, 4841, 192355, 14537643, 2135997537, 616565334097, 351243585487331, 395958973398105283, 885030941975862363649, 3928075680727698371316537, 34658158001445631936261356547, 608435501761943981290097259909211

Offset: 0

Author

Geoffrey Critzer, Oct 20 2011

Keywords

• nonn

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

Crossrefs

Cf. A053549.

Programs

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]

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

Formula

E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A053549.

Extensions

a(6), a(10) corrected by Alois P. Heinz, Mar 18 2015