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A304069 Number of simple graphs on n vertices rooted at one oriented edge. 5
0, 1, 4, 20, 120, 996, 12208, 241520, 8171936, 491317640, 53489987584, 10642774095040, 3891541970165760, 2627082058057474240, 3288629181834544457216, 7666328470407977450185984, 33415367571344085375628748800, 273361007807597539567353971109952 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A092055 A187848 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

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)