A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 4, 2, 15, 12, 3, 60, 58, 28, 4, 262, 273, 185, 55, 5, 1243, 1329, 1094, 495, 96, 6, 6358, 6839, 6293, 3757, 1148, 154, 7, 34835, 37423, 36619, 26421, 11122, 2380, 232, 8, 203307, 217606, 219931, 180482, 96454, 28975, 4518, 333, 9, 1257913, 1340597, 1376929, 1230737, 787959, 308127, 67898, 7995, 460, 10

Offset: 1

Author

Alois P. Heinz, Apr 17 2017

Keywords

• nonn
• tabl

Examples

			T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
Triangle T(n,k) begins:
      1;
      4,     2;
     15,    12,     3;
     60,    58,    28,     4;
    262,   273,   185,    55,     5;
   1243,  1329,  1094,   495,    96,    6;
   6358,  6839,  6293,  3757,  1148,  154,   7;
  34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=1-10 give: A285363, A285364, A285365, A285366, A285367, A285368, A285369, A285370, A285371, A285372.

Row sums give A000110(n) * A000217(n) = A105488(n+3).

Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).

T(2n+1,n+1) gives A285410.

Cf. A270236, A285439, A285595.

Programs

Maple

T:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
        +[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
    end:
seq(T(n), n=1..12);

Mathematica

T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)