A339045 Number of connected loopless multigraphs with n edges rooted at two noninterchangeable vertices whose removal leaves a connected graph.

1, 1, 4, 16, 69, 307, 1433, 6903, 34337, 175457, 919525, 4931233, 27023894, 151142376, 861880778, 5006906170, 29611120248, 178175786593, 1090266839041, 6781364484106, 42858210422338, 275127506187149, 1793418517202096, 11867326044069470, 79695273536227647

Offset: 1

Author

Andrew Howroyd, Nov 25 2020

Keywords

• nonn

Crossrefs

Cf. A338999, A339036, A339037, A339065.

Programs

PARI

\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g); InvEulerT(Vec(-1+G(2*n, x+A, [1,1])/(g*gr^2)))}

Formula

1/(Product_{k>=1} (1 - x^k)^a(k)) = f(x)/g(x)^2 where x*f(x) is the g.f. of A339037 and g(x) is the g.f. of A339036.