A053598 Number of n-node unlabeled digraphs without isolated nodes.

1, 0, 2, 13, 202, 9390, 1531336, 880492496, 1792477159408, 13026163465206704, 341247403996148180800, 32522568124623933138617088, 11366712907916015518547782806784, 14669074325967499043636521641422216704, 70315641946149306808455637518883828774996992

Offset: 0

Author

Vladeta Jovovic, Apr 10 2000

Keywords

• nonn
• nice

Comments

Equals first differences of A000273.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A000273, A002494, A053418 (by # arcs). Row sums of A350908.

Programs

Maple

b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
      igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
    end:
a:= n-> b(n$2, [])-`if`(n=0, 0, b(n-1$2, [])):
seq(a(n), n=0..16);  # Alois P. Heinz, Sep 04 2019

Mathematica

Needs["Combinatorica`"];
nn=15;s=Sum[NumberOfDirectedGraphs[n]x^n,{n,0,nn}];CoefficientList[Series[s (1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 09 2012 *)
Join[{1}, Table[GraphPolynomial[n, x, Directed] /. x -> 1, {n, 0, 15}] // Differences] (* Jean-François Alcover, Feb 04 2015 *)

Python

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053598(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

Formula

O.g.f.: A(x)*(1-x) where A(x) is o.g.f. for A000273. - Geoffrey Critzer, Oct 09 2012