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%I #18 Oct 20 2014 09:49:37
%S 1,3,7,1,15,3,31,7,1,63,17,3,127,43,8,1,255,108,22,3,511,273,63,9,1,
%T 1023,708,157,23,3,2047,1867,502,67,10,1,4095,4955,1562,203,26,3,8191,
%U 13256,4688,693,83,11,1,16383,35790,15533,2584,322,30,3,32767,97340
%N Irregular triangle T(n,k), n>=1, 1<=k<=ceiling(n/2), read by rows: T(n,k) is the number of different ways to select k disjoint (nonempty) subsets from {1..n} with equal element sum.
%H Alois P. Heinz, <a href="/A196231/b196231.txt">Rows n = 1..26, flattened</a>
%e T(8,4) = 3: {1,6}, {2,5}, {3,4}, {7} have element sum 7, {1,7}, {2,6}, {3,5}, {8} have element sum 8, and {1,8}, {2,7}, {3,6}, {4,5} have element sum 9.
%e Triangle begins:
%e . 1;
%e . 3;
%e . 7, 1;
%e . 15, 3;
%e . 31, 7, 1;
%e . 63, 17, 3;
%e . 127, 43, 8, 1;
%e . 255, 108, 22, 3;
%p b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j] -n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: T:= (n, k)-> add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!:
%p seq(seq(T(n, k), k=1..ceil(n/2)), n=1..15);
%t b[l_List, 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[n_, k_] := Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]}]/k!; Table[Table[T[n, k], {k, 1, Ceiling[n/2]}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Dec 17 2013, translated from Maple *)
%Y Columns k=1-10 give: A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237. Row sums give A196534. Row lengths are in A110654.
%K nonn,tabf
%O 1,2
%A _Alois P. Heinz_, Sep 29 2011
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