A304074 Number of simple connected graphs with n nodes rooted at a pair of distinguished vertices.

0, 1, 4, 23, 162, 1549, 21090, 446061, 15673518, 961338288, 105752617892, 21155707801451, 7757777336382702, 5245054939576054088, 6571185585793205495484, 15325133281701584879975433, 66813349775478836190531605234, 546646811841381587823502759339055

Offset: 1

Author

Brendan McKay, May 05 2018

Keywords

• nonn

Examples

			a(3)=4: one choice to mark two roots in the triangular graph; one choice to mark the two leaves in the linear graph; two choices to mark the center node and a leave (1st root in the center or 2nd root in the center) in the linear graph.
		

Links

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

Crossrefs

Cf. A001349 (not rooted), A303831 (vertices not distinguished), A304070 (not necessarily connected).

Cf. A000088, A000666.

Programs

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(nAndrew Howroyd, Sep 07 2019

Formula

a(n) = A304072(n) + A304073(n).

G.f.: 2*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

Extensions

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