A304072 Number of simple connected graphs with n nodes rooted at one oriented edge.

0, 1, 3, 15, 95, 848, 11043, 227978, 7915413, 482871723, 52989880632, 10588770680260, 3880844130502271, 2623179650433475894, 3285998146525888516756, 7663037181052161495721168, 33407697920116540678510839469, 273327584706334343769636571729201

Offset: 1

Author

Brendan McKay, May 05 2018

Keywords

• nonn

Examples

			a(3)=3: one choice of orienting an edge in the triangle graph; two choices of orienting an edge in the linear graph (orientation towards or away from the center node).
		

Links

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

Crossrefs

Cf. A000088, A001349 (not rooted), A304069 (not necessarily connected).

Programs

Mathematica

nmax = 20;
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[v, 2]];
a69[n_] := If[n < 2, 0, s = 0; Do[s += permcount[p]*(2^(2*Length[p] + edges[p])), {p, IntegerPartitions[n - 2]}]; s/(n - 2)!];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
gf = Sum[a69[n] x^n, {n, 0, nmax}]/Sum[a88[n] x^n, {n, 0, nmax}]+O[x]^nmax;
CoefficientList[gf, x] // Rest (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

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)}
g(n,r) = {my(s=0); forpart(p=n, s+=permcount(p)*(2^(r*#p+edges(p)))); s/n!}
seq(n)={concat([0], Vec(Ser(vector(n,n,g(n-1,2)))/Ser(vector(n,n,g(n-1,0)))))} \\ Andrew Howroyd, May 06 2018

Formula

a(n) + A304073(n) = A304074(n).

G.f.: R(x)/G(x) where R(x) is the g.f. of A304069 and G(x) is the g.f. of A000088. - Andrew Howroyd, May 06 2018

Extensions

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