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%I #19 Sep 01 2019 16:43:47
%S 1,3,13,37,134,466,1916,9409,46006,255714,1525052,9524779,58944302,
%T 355219704,2315784192,14568780212,97993669291,619342933593
%N Number of different ways to select 8 disjoint subsets from {1..n} with equal element sum.
%e a(16) = 3: {1,14}, {2,13}, {3,12}, {4,11}, {5,10}, {6,9}, {7,8}, {15} have element sum 15; {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}, {16} have element sum 16; {1,16}, {2,15}, {3,14}, {4,13}, {5,12}, {6,11}, {7,10}, {8,9} have element sum 17.
%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, 8];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 15, 25}] (* _Jean-François Alcover_, Jun 08 2018, after _Alois P. Heinz_ *)
%Y Column k=8 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196236, A196237.
%K nonn,more
%O 15,2
%A _Alois P. Heinz_, Sep 29 2011
%E a(27)-a(28) from _Alois P. Heinz_, Nov 05 2014
%E a(29)-a(32) from _Bert Dobbelaere_, Sep 01 2019
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