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