A304073 Number of simple connected graphs with n nodes rooted at one oriented non-edge.

0, 0, 1, 8, 67, 701, 10047, 218083, 7758105, 478466565, 52762737260, 10566937121191, 3876933205880431, 2621875289142578194, 3285187439267316978728, 7662096100649423384254265, 33405651855362295512020765765, 273319227135047244053866187609854

Offset

1,4

Links

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

Formula

a(n) + A304072(n) = A304074(n).

G.f.: B(x)/G(x) - (x*C(x)/G(x))^2, where B(x) is the g.f. of A304069, C(x) is the g.f. of A000666 and G(x) is the g.f. of A000088. - Andrew Howroyd, Sep 07 2019

Example

a(3)=1: no contribution from the triangle graph; one case of joining the leaves of the linear graph.
a(4)=8: we start from the 6 cases of non-oriented non-edges of A304071 and note two geometries where the orientation makes a difference: for the triangular graph with a protruding edge the orientation matters (to or from the leaf), and also for the linear graph with 4 nodes (to or from the leaf).

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)}
cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
S(n, r)={my(t=#r+1); vector(n+1, n, if(n<t, 0, my(s=0); forpart(p=n-t, s+=permcount(p)*(2^(edges(p))*(2^cross(r, p)))); s/(n-t)!))}
seq(n)={my(g=Ser(S(n, []))); Vec(Ser(S(n, [1, 1]))/g - (Ser(S(n, [1]))/g)^2, -n)} \\ Andrew Howroyd, Sep 07 2019

Crossrefs

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

Cf. A000088, A000666.

Sequence in context: A152055 A363309 A000434 * A250258 A192091 A050841

Adjacent sequences: A304070 A304071 A304072 * A304074 A304075 A304076

Keyword

nonn

Author

Brendan McKay, May 05 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 07 2019

Status

approved