%I #22 Sep 02 2019 14:39:37
%S 1,3,12,33,114,403,1618,8946,45917,189428,979841,5497818,31708309,
%T 178006222,1091681487,6207647636,32636979255,184162388392,
%U 1069147827024,6446977283374
%N Number of different ways to select 7 disjoint subsets from {1..n} with equal element sum.
%e a(14) = 3:
%e {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13} have element sum 13; {1,13}, {2,12}, {3,11}, {4,10}, {5,9}, {6,8}, {14} have element sum 14; {1,14}, {2,13}, {3,12}, {4,11}, {5,10}, {6,9}, {7,8} have element sum 15.
%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, 7];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 13, 25}] (* _Jean-François Alcover_, Jun 08 2018, after _Alois P. Heinz_ *)
%Y Column k=7 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196235, A196236, A196237.
%K nonn,more
%O 13,2
%A _Alois P. Heinz_, Sep 29 2011
%E a(26)-a(28) from _Alois P. Heinz_, Sep 26 2014
%E a(29)-a(32) from _Bert Dobbelaere_, Sep 02 2019