A307432 Number T(n,k) of partitions of n into parts whose bitwise AND equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 3, 2, 1, 0, 0, 1, 5, 3, 1, 1, 0, 0, 1, 9, 4, 1, 0, 0, 0, 0, 1, 11, 6, 3, 0, 1, 0, 0, 0, 1, 18, 6, 3, 1, 1, 0, 0, 0, 0, 1, 27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1, 38, 11, 4, 0, 2, 0, 0, 0, 0, 0, 0, 1, 53, 13, 6, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1

Offset

0,12

Links

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

Wikipedia, Bitwise operation

Wikipedia, Partition (number theory)

Example

T(6,0) = 5: 11112, 1122, 123, 114, 24.
T(6,1) = 3: 111111, 1113, 15.
T(6,2) = 1: 222.
T(6,3) = 1: 33.
T(6,6) = 1: 6.
Triangle T(n,k) begins:
   1;
   0, 1;
   0, 1, 1;
   1, 1, 0, 1;
   1, 2, 1, 0, 1;
   3, 2, 1, 0, 0, 1;
   5, 3, 1, 1, 0, 0, 1;
   9, 4, 1, 0, 0, 0, 0, 1;
  11, 6, 3, 0, 1, 0, 0, 0, 1;
  18, 6, 3, 1, 1, 0, 0, 0, 0, 1;
  27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1;
  ...

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[And](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(
         b(n$2, `if`(n=0, 0, 2^ilog2(2*n)-1))):
seq(T(n), n=0..14);

Crossrefs

Column k=0 gives A307435.

Row sums give A000041.

Main diagonal gives A000012.

Cf. A050314 (the same for XOR), A307431 (the same for OR).

Sequence in context: A048983 A301505 A301504 * A256140 A321391 A244003

Adjacent sequences: A307429 A307430 A307431 * A307433 A307434 A307435

Keyword

nonn,tabl,look,base

Author

Alois P. Heinz, Apr 08 2019

Status

approved