A307505 Number T(n,k) of partitions of n into distinct parts whose bitwise XOR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 0, 0, 0, 5, 1, 0, 5, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2

Offset: 0

Author

Alois P. Heinz, Apr 11 2019

Keywords

• nonn
• tabl
• look
• base

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 2;
  0, 0, 1, 0, 1;
  0, 1, 0, 0, 0, 2;
  1, 0, 0, 0, 1, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 5;
  0, 0, 0, 0, 1, 0, 4, 0, 1;
  0, 1, 0, 0, 0, 4, 0, 1, 0, 2;
  1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..360, flattened
• Wikipedia, Bitwise operation
• Wikipedia, Partition (number theory)

Crossrefs

Bisection (even part) of column k=0 gives A307506.

Row sums give A000009.

Main diagonal gives A050315.

Cf. A050314.

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-1), Bits[Xor](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);

Formula

T(n,k) = 0 if n+k is odd.