A307431 Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 1, 0, 4, 0, 2, 0, 1, 1, 5, 0, 2, 2, 0, 1, 0, 7, 0, 2, 0, 5, 0, 1, 1, 8, 1, 2, 2, 6, 1, 0, 1, 0, 11, 0, 4, 0, 12, 0, 2, 0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2, 0, 1, 0, 15, 0, 5, 0, 28, 0, 2, 0, 5, 0, 1, 1, 17, 1, 5, 5, 35, 0, 2, 2, 6, 2

Offset: 0

Author

Alois P. Heinz, Apr 08 2019

Keywords

• nonn
• tabl
• look
• base

Examples

			T(6,1) = 1: 111111.
T(6,2) = 1: 222.
T(6,3) = 5: 11112, 1122, 1113, 123, 33.
T(6,5) = 2: 114, 15.
T(6,6) = 2: 24, 6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  2;
  0, 1, 1,  2, 1;
  0, 1, 0,  4, 0, 2;
  0, 1, 1,  5, 0, 2, 2;
  0, 1, 0,  7, 0, 2, 0,  5;
  0, 1, 1,  8, 1, 2, 2,  6, 1;
  0, 1, 0, 11, 0, 4, 0, 12, 0, 2;
  0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Bitwise operation
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-1 give: A000007, A057427.

Row sums give: A000041.

Main diagonal gives A050315.

Cf. A050314 (the same for XOR), A307432 (the same for AND).

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), Bits[Or](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);