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

Author

Christian G. Bower, Sep 15 1999

Keywords

• nonn
• tabl
• base
• look

Examples

			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;
  ...
		

Links

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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)