More terms from Ronald R. C. Read (rcread(AT)math.uwaterloo.ca) and Vladeta Jovovic.

OEIS Server: https://oeis.org/edit/global/2963

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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_] := 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]}];

a[n_] := (s = 0; Do[s += permcount[2 p]*2^edges[p]*If[OddQ[n], n *4^Length[p], 1], {p, IntegerPartitions[n/2 // Floor]}]; s/n!);

Array[a, 16] (* Jean-François Alcover, Aug 26 2019, after Andrew Howroyd *)

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Andrew Howroyd, <a href="/A003086/b003086.txt">Table of n, a(n) for n = 1..80</a>

(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) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, 2*v[i]-1)}

a(n) = {my(s=0); forpart(p=n\2, s+=permcount(2*Vec(p))*2^edges(p)*if(n%2, n*4^#p, 1)); s/n!} \\ Andrew Howroyd, Sep 16 2018

Cf. A000171, A002785.

_N. J. A. Sloane_._

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More terms from R. C. Read (rcread(AT)math.uwaterloo.ca) and _Vladeta Jovovic (vladeta(AT)eunet.rs)._.

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