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W. Oberschelp, <a href="http://gdz.sub.uni-goettingen.de/dms/load/imgresolveppn/?PIDPPN=GDZPPN002298732">Kombinatorische Anzahlbestimmungen in Relationen</a>, Math. Ann., 174 (1967), 53-78.

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Michel Marcus: better url

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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[p_] := Sum[Ceiling[(p[[i]] - 1)*((p[[i]] - 2)/6)], {i, 1, Length[p]}] + Sum[Sum[c = p[[i]]; d = p[[j]]; GCD[c, d]*(c + d - 2 + Mod[(c - d)/GCD[c, d], 2])/2 + Sum[c*d*p[[k]]/LCM[c, d, p[[k]]], {k, 1, j - 1}], {j, 1, i - 1}], {i, 2, Length[p]}];

a[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];

a /@ Range[0, 12] (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)

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Alois P. Heinz, <a href="/A000665/b000665_1.txt">Table of n, a(n) for n = 0..28</a> (first 26 terms from Andrew Howroyd)

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OEIS Server: Installed new b-file as b000665.txt.  Old b-file is now b000665_1.txt.

AndrewAlois HowroydP. Heinz, <a href="/A000665/b000665_1.txt">Table of n, a(n) for n = 0..2528</a>> (first 26 terms from Andrew Howroyd)

Column k=3 of A309858.

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