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%I #19 Sep 05 2019 02:29:36
%S 1,3,10,26,83,322,1182,3971,15662,69371,328016,1460297,6080910,
%T 26901643,123926071,598722099,2838432721,13220493552,63710261040,
%U 312134646974,1554373859464,7673048166979,37597940705361,186986406578372
%N Number of different ways to select 5 disjoint subsets from {1..n} with equal element sum.
%e a(10) = 3: {1,8}, {2,7}, {3,6}, {4,5}, {9} have element sum 9; {1,9}, {2,8}, {3,7}, {4,6}, {10} have element sum 10; {1,10}, {2,9}, {3,8}, {4,7}, {5,6} have element sum 11.
%t b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}] ]];
%t T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
%t a[n_] := T[n, 5];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 9, 25}] (* _Jean-François Alcover_, Jun 08 2018, after _Alois P. Heinz_ *)
%Y Column k=5 of A196231. Cf. A000225, A161943, A164934, A164949, A196233, A196234, A196235, A196236, A196237.
%K nonn,more
%O 9,2
%A _Alois P. Heinz_, Sep 29 2011
%E a(26)-a(28) from _Alois P. Heinz_, Sep 25 2014
%E a(29)-a(32) from _Bert Dobbelaere_, Sep 05 2019
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