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A304069 Number of simple graphs on n vertices rooted at one oriented edge.

%I A304069 #15 Jul 03 2018 12:15:31
%S A304069 0,1,4,20,120,996,12208,241520,8171936,491317640,53489987584,
%T A304069 10642774095040,3891541970165760,2627082058057474240,
%U A304069 3288629181834544457216,7666328470407977450185984,33415367571344085375628748800,273361007807597539567353971109952
%N A304069 Number of simple graphs on n vertices rooted at one oriented edge.
%C A304069 This is also the number of simple graphs rooted at an oriented non-edge.
%C A304069 The graphs do not need to be connected here; see A304072 for the connected graphs.
%H A304069 Andrew Howroyd, <a href="/A304069/b304069.txt">Table of n, a(n) for n = 1..50</a>
%F A304069 2*a(n) = A304070(n).
%e A304069 a(3)=4: no contribution from the graph with 3 isolated nodes. 1 case of the connected graph with 2 nodes and an isolated node. 2 cases of the linear graph with 3 nodes (orientation either towards or away from the middle node). 1 case of the triangular graph.
%t A304069 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A304069 edges[v_] := Sum[GCD[v[[i]], v[[j]] ], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[#, 2]& /@ v];
%t A304069 a[n_] := If[n < 2, 0, s = 0; Do[s += permcount[p]*(2^(2*Length[p] + edges[p])), {p, IntegerPartitions[n - 2]}]; s/(n - 2)!];
%t A304069 Array[a, 18] (* _Jean-François Alcover_, Jul 03 2018, after _Andrew Howroyd_ *)
%o A304069 (PARI)
%o A304069 permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
%o A304069 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2)}
%o A304069 a(n)= {if(n<2, 0, my(s=0); forpart(p=n-2, s+=permcount(p)*(2^(2*#p+edges(p)))); s/(n-2)!)} \\ _Andrew Howroyd_, May 06 2018
%Y A304069 Cf. A000088 (not rooted).
%K A304069 nonn
%O A304069 1,3
%A A304069 _Brendan McKay_, May 05 2018
%E A304069 Terms a(13) and beyond from _Andrew Howroyd_, May 06 2018