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

0,10

Links

Alois P. Heinz, Rows n = 0..200, flattened

Wikipedia, Bitwise operation

Wikipedia, Partition (number theory)

Example

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;
  ...

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);

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).

Sequence in context: A155041 A334296 A227003 * A263848 A348157 A108455

Adjacent sequences: A307428 A307429 A307430 * A307432 A307433 A307434

Keyword

nonn,tabl,look,base

Author

Alois P. Heinz, Apr 08 2019

Status

approved