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

0, 1, 4, 20, 120, 996, 12208, 241520, 8171936, 491317640, 53489987584, 10642774095040, 3891541970165760, 2627082058057474240, 3288629181834544457216, 7666328470407977450185984, 33415367571344085375628748800, 273361007807597539567353971109952

Offset

1,3

Comments

This is also the number of simple graphs rooted at an oriented non-edge.

The graphs do not need to be connected here; see A304072 for the connected graphs.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Formula

2*a(n) = A304070(n).

Example

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.

Mathematica

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];
edges[v_] := Sum[GCD[v[[i]], v[[j]] ], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[#, 2]& /@ v];
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)!];
Array[a, 18] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

Prog

(PARI)
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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
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

Crossrefs

Cf. A000088 (not rooted).

Sequence in context: A187848 A370653 A001715 * A020028 A020118 A009351

Adjacent sequences: A304066 A304067 A304068 * A304070 A304071 A304072

Keyword

nonn

Author

Brendan McKay, May 05 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, May 06 2018

Status

approved