A274537 Number T(n,k) of set partitions of [n] into k blocks such that each element is contained in a block whose index parity coincides with the parity of the element; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 3, 7, 2, 1, 0, 0, 1, 7, 14, 13, 3, 1, 0, 0, 1, 7, 35, 26, 22, 3, 1, 0, 0, 1, 15, 70, 113, 66, 34, 4, 1, 0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1, 0, 0, 1, 31, 310, 833, 933, 719, 200, 70, 5, 1

Offset

0,19

Comments

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

Formula

Sum_{k=0..n} k * T(n,k) = A364267(n). - Alois P. Heinz, Jul 16 2023

Example

T(6,2) = 1: 135|246.
T(6,3) = 3: 13|246|5, 15|246|3, 1|246|35.
T(6,4) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
T(6,5) = 2: 1|26|3|4|5, 1|2|3|46|5.
T(6,6) = 1: 1|2|3|4|5|6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  1,   1;
  0, 0, 1,  3,   2,   1;
  0, 0, 1,  3,   7,   2,   1;
  0, 0, 1,  7,  14,  13,   3,   1;
  0, 0, 1,  7,  35,  26,  22,   3,  1;
  0, 0, 1, 15,  70, 113,  66,  34,  4, 1;
  0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1;
  ...

Maple

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

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n==0, x^m, Sum[If[Mod[j, 2]==t, b[n-1, Max[m, j], 1-t], 0], {j, 1, m+1}]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Crossrefs

Row sums give A274538.

Columns k=0-10 give: A000007, A000007(n-1), A000012(n-2), A052551(n-3), A274868, A274869, A274870, A274871, A274872, A274873, A274874.

T(2n,n) gives A274875.

Main diagonal and lower diagonals give: A000012, A004526, A002623(n-2) or A173196.

Cf. A364267.

Sequence in context: A330959 A083199 A327187 * A305234 A021761 A221960

Adjacent sequences: A274534 A274535 A274536 * A274538 A274539 A274540

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 27 2016

Status

approved