A307432 - OEIS

%I #19 Apr 08 2019 17:28:37

%S 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,

%T 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,

%U 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

%N 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.

%H Alois P. Heinz, <a href="/A307432/b307432.txt">Rows n = 0..200, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%e T(6,0) = 5: 11112, 1122, 123, 114, 24.

%e T(6,1) = 3: 111111, 1113, 15.

%e T(6,2) = 1: 222.

%e T(6,3) = 1: 33.

%e T(6,6) = 1: 6.

%e Triangle T(n,k) begins:

%e    1;

%e    0, 1;

%e    0, 1, 1;

%e    1, 1, 0, 1;

%e    1, 2, 1, 0, 1;

%e    3, 2, 1, 0, 0, 1;

%e    5, 3, 1, 1, 0, 0, 1;

%e    9, 4, 1, 0, 0, 0, 0, 1;

%e   11, 6, 3, 0, 1, 0, 0, 0, 1;

%e   18, 6, 3, 1, 1, 0, 0, 0, 0, 1;

%e   27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1;

%e   ...

%p b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,

%p       b(n, i-1, k)+b(n-i, min(n-i, i), Bits[And](i, k))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(

%p          b(n$2, `if`(n=0, 0, 2^ilog2(2*n)-1))):

%p seq(T(n), n=0..14);

%Y Column k=0 gives A307435.

%Y Row sums give A000041.

%Y Main diagonal gives A000012.

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

%K nonn,tabl,look,base

%O 0,12

%A _Alois P. Heinz_, Apr 08 2019