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

0,2

Links

Table of n, a(n) for n=0..25.

Example

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

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 *)

Prog

(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])))}

Crossrefs

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

Sequence in context: A344217 A033122 A330246 * A005511 A349695 A003127

Adjacent sequences: A339062 A339063 A339064 * A339066 A339067 A339068

Keyword

nonn

Author

Andrew Howroyd, Nov 22 2020

Status

approved