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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.

%I A307431 #21 Apr 08 2019 17:29:08
%S A307431 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,
%T A307431 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,
%U A307431 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
%N 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.
%H A307431 Alois P. Heinz, <a href="/A307431/b307431.txt">Rows n = 0..200, flattened</a>
%H A307431 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>
%H A307431 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A307431 T(6,1) = 1: 111111.
%e A307431 T(6,2) = 1: 222.
%e A307431 T(6,3) = 5: 11112, 1122, 1113, 123, 33.
%e A307431 T(6,5) = 2: 114, 15.
%e A307431 T(6,6) = 2: 24, 6.
%e A307431 Triangle T(n,k) begins:
%e A307431   1;
%e A307431   0, 1;
%e A307431   0, 1, 1;
%e A307431   0, 1, 0,  2;
%e A307431   0, 1, 1,  2, 1;
%e A307431   0, 1, 0,  4, 0, 2;
%e A307431   0, 1, 1,  5, 0, 2, 2;
%e A307431   0, 1, 0,  7, 0, 2, 0,  5;
%e A307431   0, 1, 1,  8, 1, 2, 2,  6, 1;
%e A307431   0, 1, 0, 11, 0, 4, 0, 12, 0, 2;
%e A307431   0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2;
%e A307431   ...
%p A307431 b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
%p A307431       b(n, i-1, k)+b(n-i, min(n-i, i), Bits[Or](i, k))))
%p A307431     end:
%p A307431 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A307431 seq(T(n), n=0..14);
%Y A307431 Columns k=0-1 give: A000007, A057427.
%Y A307431 Row sums give: A000041.
%Y A307431 Main diagonal gives A050315.
%Y A307431 Cf. A050314 (the same for XOR), A307432 (the same for AND).
%K A307431 nonn,tabl,look,base
%O A307431 0,10
%A A307431 _Alois P. Heinz_, Apr 08 2019