A270702 Total sum T(n,k) of the sizes of all blocks with minimal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 9, 4, 2, 30, 16, 9, 5, 112, 67, 41, 25, 15, 463, 299, 195, 127, 82, 52, 2095, 1429, 979, 670, 456, 307, 203, 10279, 7307, 5204, 3702, 2623, 1845, 1283, 877, 54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140, 306298, 230884, 174029, 131007, 98367, 73561, 54692, 40338, 29427, 21147

Offset

1,2

Links

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

Wikipedia, Partition of a set

Formula

T(n,k) = A270701(n,n-k+1).

Example

Row n=3 is [9, 4, 2] = [3+2+2+1+1, 0+0+1+2+1, 0+1+0+0+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
Triangle T(n,k) begins:
:      1;
:      3,     1;
:      9,     4,     2;
:     30,    16,     9,     5;
:    112,    67,    41,    25,    15;
:    463,   299,   195,   127,    82,    52;
:   2095,  1429,   979,   670,   456,   307,  203;
:  10279,  7307,  5204,  3702,  2623,  1845, 1283,  877;
:  54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140;

Maple

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
     `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
      [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
     `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
    end:
T:= (n, k)-> b(n, 0, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12);

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m + 1, 0, Function[p, p + If[j == -t || t == 1 && j == m + 1, {0, p[[1]]}, 0] ][b[n - 1, Max[m, j], If[t == 1 && j == m + 1, -j, If[t < 0, t, If[t > 0, t - 1, 0]]]]]], {j, 1, m + 1}]];
T[n_, k_] := b[n, 0, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *)

Crossrefs

Columns k=1-10 give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.

Main and lower diagonals give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.

Row sums give A070071.

Reflected triangle gives A270701.

T(2n-1,n) gives A270703.

Sequence in context: A187887 A016577 A308704 * A124573 A127550 A021317

Adjacent sequences: A270699 A270700 A270701 * A270703 A270704 A270705

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 21 2016

Status

approved