A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

1, 2, 3, 5, 8, 9, 15, 25, 29, 31, 52, 89, 106, 115, 120, 203, 354, 431, 474, 499, 514, 877, 1551, 1923, 2141, 2273, 2355, 2407, 4140, 7403, 9318, 10489, 11224, 11695, 12002, 12205, 21147, 38154, 48635, 55286, 59595, 62434, 64331, 65614, 66491, 115975, 210803, 271617, 311469, 338019, 355951, 368205, 376665, 382559, 386699

Offset

1,2

Links

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

Formula

T(n,k) = Sum_{i=1..k} A120058(n,i)*B(n-i+1), where B(n) are the Bell numbers, (A000110).

Example

The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9.
Table starts:
   1;
   2,  3;
   5,  8,   9;
  15, 25,  29,  31;
  52, 89, 106, 115, 120;
  ...

Maple

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
      add((p-> [p[1], expand(p[2]*x+p[1]*j)])(
        b(n-1, max(m, j))), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]):
seq(T(n), n=1..10);  # Alois P. Heinz, Mar 24 2016

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 08 2016, after Alois P. Heinz *)

Crossrefs

Cf. A120058, A120095. First column is A000110.

Main diagonal is A087648(n-1).

Row sums give A346772.

Sequence in context: A295082 A168154 A278334 * A099422 A294913 A056903

Adjacent sequences: A120054 A120055 A120056 * A120058 A120059 A120060

Keyword

nonn,tabl

Author

Franklin T. Adams-Watters, Jun 06 2006, Jun 07 2006

Status

approved