cp's OEIS Frontend

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

%I A304074 #16 Sep 08 2019 00:59:06
%S A304074 0,1,4,23,162,1549,21090,446061,15673518,961338288,105752617892,
%T A304074 21155707801451,7757777336382702,5245054939576054088,
%U A304074 6571185585793205495484,15325133281701584879975433,66813349775478836190531605234,546646811841381587823502759339055
%N A304074 Number of simple connected graphs with n nodes rooted at a pair of distinguished vertices.
%H A304074 Andrew Howroyd, <a href="/A304074/b304074.txt">Table of n, a(n) for n = 1..50</a>
%F A304074 a(n) = A304072(n) + A304073(n).
%F A304074 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
%e A304074 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.
%o A304074 (PARI)
%o A304074 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}
%o A304074 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o A304074 cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
%o A304074 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)!))}
%o A304074 seq(n)={my(g=Ser(S(n,[]))); Vec(2*Ser(S(n,[1,1]))/g - (Ser(S(n,[1]))/g)^2, -n)} \\ _Andrew Howroyd_, Sep 07 2019
%Y A304074 Cf. A001349 (not rooted), A303831 (vertices not distinguished), A304070 (not necessarily connected).
%Y A304074 Cf. A000088, A000666.
%K A304074 nonn
%O A304074 1,3
%A A304074 _Brendan McKay_, May 05 2018
%E A304074 Terms a(13) and beyond from _Andrew Howroyd_, Sep 07 2019