A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

1, 1, 3, 11, 43, 180, 804, 3763, 18331, 92330, 478795, 2547885, 13880832, 77284220, 439146427, 2543931619, 15010717722, 90154755356, 550817917537, 3421683388385, 21601986281226, 138548772267326, 902439162209914, 5967669851051612, 40053432076016812

Offset: 1

Author

Rainer Rosenthal, Nov 18 2020

Keywords

• nonn

Comments

This sequence counts the CDE-descendants of a single edge A-Z.

[C]onnect: different nodes {P,Q} != {A,Z} may form a new edge P-Q.

[D]issect: any edge P-Q may be dissected into P-M-Q with a new node M.

[E]xtend: any node P not in {A,Z} may form a new edge P-Q with a new node Q.

These basic operations were motivated by A338487, which seemed to count the CDE-descendants of K_4 with edge A-Z removed.

Examples

			The a(3) = 3 CDE-descendants of A-Z with 3 edges are
.
         A          A          A
        ( )        /          /
         o        o - o      o - o
         |           /        \
         Z          Z          Z
.
        DCC        DD         DE
.
		

References

• Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Links

• Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23, 2004.

Crossrefs

Cf. A180414, A337517, A338487, A339038, A339045, 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, u=InvEulerT(Vec(-1+G(2*n, x+A,[1,1])/(g*gr^2))), t=InvEulerT(Vec(-1+G(2*n, x+A,[2])/(g*subst(gr,x,x^2)))), v=vector(n)); for(n=1, #v, v[n]=(u[n]+t[n]-if(n%2==0,u[n/2]-v[n/2]))/2); v} \\ Andrew Howroyd, Nov 20 2020

Extensions

a(7)-a(25) from Andrew Howroyd, Nov 20 2020