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

1,2

Links

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Partition of a set

Example

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;
  ...

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 *)

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.

Sequence in context: A299789 A121662 A130042 * A109922 A090640 A302213

Adjacent sequences: A285359 A285360 A285361 * A285363 A285364 A285365

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 17 2017

Status

approved