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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.

%I A307505 #18 Apr 11 2019 11:59:02
%S A307505 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,
%T A307505 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,
%U A307505 0,4,0,2,0,1,0,0,0,5,1,0,5,0,0,0,2,0,1,0,4,0,2
%N 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.
%H A307505 Alois P. Heinz, <a href="/A307505/b307505.txt">Rows n = 0..360, flattened</a>
%H A307505 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>
%H A307505 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A307505 T(n,k) = 0 if n+k is odd.
%e A307505 Triangle T(n,k) begins:
%e A307505   1;
%e A307505   0, 1;
%e A307505   0, 0, 1;
%e A307505   0, 0, 0, 2;
%e A307505   0, 0, 1, 0, 1;
%e A307505   0, 1, 0, 0, 0, 2;
%e A307505   1, 0, 0, 0, 1, 0, 2;
%e A307505   0, 0, 0, 0, 0, 0, 0, 5;
%e A307505   0, 0, 0, 0, 1, 0, 4, 0, 1;
%e A307505   0, 1, 0, 0, 0, 4, 0, 1, 0, 2;
%e A307505   1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2;
%e A307505   ...
%p A307505 b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
%p A307505       b(n, i-1, k)+b(n-i, min(n-i, i-1), Bits[Xor](i, k))))
%p A307505     end:
%p A307505 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A307505 seq(T(n), n=0..14);
%Y A307505 Bisection (even part) of column k=0 gives A307506.
%Y A307505 Row sums give A000009.
%Y A307505 Main diagonal gives A050315.
%Y A307505 Cf. A050314.
%K A307505 nonn,tabl,look,base
%O A307505 0,10
%A A307505 _Alois P. Heinz_, Apr 11 2019