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

%I A053598 #26 Jul 05 2024 18:04:56
%S A053598 1,0,2,13,202,9390,1531336,880492496,1792477159408,13026163465206704,
%T A053598 341247403996148180800,32522568124623933138617088,
%U A053598 11366712907916015518547782806784,14669074325967499043636521641422216704,70315641946149306808455637518883828774996992
%N A053598 Number of n-node unlabeled digraphs without isolated nodes.
%C A053598 Equals first differences of A000273.
%H A053598 Alois P. Heinz, <a href="/A053598/b053598.txt">Table of n, a(n) for n = 0..60</a>
%F A053598 O.g.f.: A(x)*(1-x) where A(x) is o.g.f. for A000273. - _Geoffrey Critzer_, Oct 09 2012
%p A053598 b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
%p A053598       igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
%p A053598       add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
%p A053598     end:
%p A053598 a:= n-> b(n$2, [])-`if`(n=0, 0, b(n-1$2, [])):
%p A053598 seq(a(n), n=0..16);  # _Alois P. Heinz_, Sep 04 2019
%t A053598 Needs["Combinatorica`"];
%t A053598 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 *)
%t A053598 Join[{1}, Table[GraphPolynomial[n, x, Directed] /. x -> 1, {n, 0, 15}] // Differences] (* _Jean-François Alcover_, Feb 04 2015 *)
%o A053598 (Python)
%o A053598 from itertools import combinations
%o A053598 from math import prod, factorial, gcd
%o A053598 from fractions import Fraction
%o A053598 from sympy.utilities.iterables import partitions
%o A053598 def A053598(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r,s)<<1 for r,s in combinations(p.keys(),2))+sum(r*(q*r-1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))-int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r,s)<<1 for r,s in combinations(p.keys(),2))+sum(r*(q*r-1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n-1))) if n else 1 # _Chai Wah Wu_, Jul 05 2024
%Y A053598 Cf. A000273, A002494, A053418 (by # arcs). Row sums of A350908.
%K A053598 nonn,nice
%O A053598 0,3
%A A053598 _Vladeta Jovovic_, Apr 10 2000