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MATHEMATICA
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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_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
b[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
edges4[v_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[2 v[[i]] - 1, {i, 1, Length[v]}];
c[n_] := (s = 0; Do[s += permcount[2 p]*2^edges4[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);
a[n_] := (b[n] + c[n])/2;
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