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

0,10

Links

Alois P. Heinz, Rows n = 0..360, flattened

Wikipedia, Bitwise operation

Wikipedia, Partition (number theory)

Formula

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

Example

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;
  ...

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);

Crossrefs

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

Row sums give A000009.

Main diagonal gives A050315.

Cf. A050314.

Sequence in context: A340999 A119395 A087476 * A035162 A121454 A025462

Adjacent sequences: A307502 A307503 A307504 * A307506 A307507 A307508

Keyword

nonn,tabl,look,base

Author

Alois P. Heinz, Apr 11 2019

Status

approved