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%I M3717 N1519 #33 Jun 17 2022 14:45:49
%S 4,136,44224,179228736,9383939974144,6558936236286040064,
%T 62879572771326489528942592,8439543710699844562674685252214784,
%U 16110027001555070629022725866559372785352704,442829046878106126159584032189649757399796014050181120
%N Number of relational systems on n nodes. Also number of directed 3-multigraphs with loops on n nodes.
%D W. Oberschelp, "Strukturzahlen in endlichen Relationssystemen", in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A001374/b001374.txt">Table of n, a(n) for n = 1..40</a>
%H W. Oberschelp, <a href="/A000662/a000662.pdf"> Strukturzahlen in endlichen Relationssystemen</a>, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]
%t 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];
%t edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
%t a[n_] := (s=0; Do[s += permcount[p]*4^edges[p], {p, IntegerPartitions[n]}]; s/n!);
%t Array[a, 15] (* _Jean-François Alcover_, Jul 08 2018, after _Andrew Howroyd_ *)
%o (PARI)
%o 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}
%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i])}
%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*4^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 22 2017
%Y Cf. A000595, A004105, A053468, A053516.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jan 14 2000
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