A307432 -id:A307432 - cp's OEIS frontend

A050314 Triangle: a(n,k) = number of partitions of n whose xor-sum is k.

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 1, 0, 3, 0, 2, 0, 2, 4, 0, 3, 0, 2, 0, 2, 0, 4, 0, 4, 0, 2, 0, 5, 6, 0, 5, 0, 4, 0, 6, 0, 1, 0, 8, 0, 6, 0, 8, 0, 6, 0, 2, 10, 0, 9, 0, 11, 0, 8, 0, 2, 0, 2, 0, 11, 0, 14, 0, 12, 0, 12, 0, 2, 0, 5, 16, 0, 18, 0, 15, 0, 16, 0, 4, 0, 6, 0, 2, 0, 23, 0, 20, 0, 20, 0, 19, 0, 8, 0, 6, 0, 5

Offset: 0

Christian G. Bower, Sep 15 1999

			Triangle: a(n,k) begins:
   1;
   0,  1;
   1,  0,  1;
   0,  1,  0,  2;
   2,  0,  2,  0,  1;
   0,  3,  0,  2,  0,  2;
   4,  0,  3,  0,  2,  0,  2;
   0,  4,  0,  4,  0,  2,  0,  5;
   6,  0,  5,  0,  4,  0,  6,  0, 1;
   0,  8,  0,  6,  0,  8,  0,  6, 0, 2;
  10,  0,  9,  0, 11,  0,  8,  0, 2, 0, 2;
   0, 11,  0, 14,  0, 12,  0, 12, 0, 2, 0, 5;
  16,  0, 18,  0, 15,  0, 16,  0, 4, 0, 6, 0, 2;
  ...

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

a(2n,0) = A048833(n). a(2n+1,1) = A050316(n). a(n,n) = A050315(n).
Row sums give A000041.
a(4n,2n) gives A370874.
Cf. A307431, A307432.

with(Bits):
b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      add(b(n-i*j, i-1, `if`(j::even, k, Xor(i, k))), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Dec 01 2015

b[n_, i_, k_] := b[n, i, k] = If[n==0, x^k, If[i<1, 0, Sum[b[n-i*j, i-1, If[EvenQ[j], k, BitXor[i, k]]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 24 2016, after Alois P. Heinz *)

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.

Original entry on oeis.org

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

Alois P. Heinz, Apr 08 2019

			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;
  ...

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

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).

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);

A307435 Number of partitions of n into parts whose bitwise AND equals 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 5, 9, 11, 18, 27, 38, 53, 75, 102, 137, 178, 238, 313, 406, 528, 677, 865, 1093, 1382, 1742, 2181, 2717, 3377, 4175, 5146, 6320, 7737, 9454, 11516, 13986, 16950, 20473, 24682, 29672, 35631, 42663, 50992, 60807, 72399, 86008, 102027, 120793

Offset: 0

Alois P. Heinz, Apr 08 2019

			a(0) = 1: the empty partition.
a(3) = 1: 21.
a(4) = 1: 211.
a(5) = 3: 2111, 221, 41.
a(6) = 5: 21111, 2211, 321, 411, 42.
a(7) = 9: 211111, 22111, 2221, 3211, 4111, 421, 43, 52, 61.
a(8) = 11: 2111111, 221111, 22211, 32111, 3221, 41111, 4211, 422, 431, 521, 611.
a(9) = 18: 21111111, 2211111, 222111, 22221, 321111, 32211, 3321, 411111, 42111, 4221, 4311, 432, 441, 5211, 522, 6111, 621, 81.

Alois P. Heinz, Table of n, a(n) for n = 0..1024
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Column k=0 of A307432.

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, k)+b(n-i, min(n-i, i), Bits[And](i, k))))
    end:
a:= n-> b(n$2, `if`(n=0, 0, 2^ilog2(2*n)-1)):
seq(a(n), n=0..50);

cp's OEIS Frontend

A050314 Triangle: a(n,k) = number of partitions of n whose xor-sum is k.

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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.

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A307435 Number of partitions of n into parts whose bitwise AND equals 0.

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