A339065 Number of unlabeled loopless multigraphs with n edges rooted at two noninterchangeable vertices.

1, 4, 17, 69, 281, 1147, 4784, 20345, 88726, 396971, 1823920, 8605364, 41684417, 207201343, 1056244832, 5518054182, 29521703655, 161625956908, 904857279576, 5176569819167, 30241443710950, 180293374961036, 1096240011165724, 6793998104717138, 42894087222036022, 275735424352928682

Offset: 0

Author

Andrew Howroyd, Nov 22 2020

Keywords

• nonn

Examples

			The a(1) = 4 cases correspond to a single edge which can be attached to zero, one or both of the roots.
		

Crossrefs

Cf. A050535, A007717 (one root), A339042, A339063, A339066.

Programs

Mathematica

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[With[{g = GCD[v[[i]], v[[j]]]}, t[v[[i]]*v[[j]]/ g]^g], {i, 2, Length[v]}, {j, 1, i - 1}]*Product[With[{c = v[[i]]}, t[c]^Quotient[c-1, 2]*If[OddQ[c], 1, t[c/2]]], {i, 1, Length[v]}];
G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 1/(1 - x^#) &], {p, IntegerPartitions[n]}]; s/n!];
seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]//CoefficientList[#, x]&];
seq[15] (* Jean-François Alcover, Dec 01 2020, after Andrew Howroyd *)

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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x, r)={my(s=0); forpart(p=n, s+=permcount(p)*edges(concat(r, Vec(p)), i->1/(1-x^i))); s/n!}
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}