A197927 The number of isolated nodes in all labeled directed graphs (with self loops allowed) on n nodes.

0, 1, 4, 48, 2048, 327680, 201326592, 481036337152, 4503599627370496, 166020696663385964544, 24178516392292583494123520, 13944156602510523416463735259136, 31901471898837980949691369446728269824, 289909687580898100839964337544428699577745408

Offset: 0

Author

Geoffrey Critzer, Oct 19 2011

Keywords

• nonn

Comments

Here, isolated means indegree = outdegree = 0.

a(n) is also the number of directed graphs on [n] (no self loops allowed, A053763) with a distinguished vertex of indegree 0. - Geoffrey Critzer, Apr 01 2023

Crossrefs

Cf. A011266, A053763, A002416.

Programs

Mathematica

a = Sum[2^(n^2)x^n/n!, {n,0,20}]; Range[0,12]! CoefficientList[Series[x a, {x,0,12}], x]

Formula

E.g.f.: x*A(x) where A(x) = Sum_{n>=0} 2^(n^2)*x^n/n!.

a(n) = n * 2^((n-1)^2) = n*A002416(n-1).

Sum_{n>=0} a(n)*z^n/B(n) = z*Sum_{n>=0} A053763(n)*z^n/B(n) where B(n) = n!*2^binomial(n,2). - Geoffrey Critzer, Apr 01 2023