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A307432 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.

%I A307432 #19 Apr 08 2019 17:28:37
%S A307432 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 A307432 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 A307432 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 A307432 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 A307432 Alois P. Heinz, <a href="/A307432/b307432.txt">Rows n = 0..200, flattened</a>
%H A307432 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise operation">Bitwise operation</a>
%H A307432 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A307432 T(6,0) = 5: 11112, 1122, 123, 114, 24.
%e A307432 T(6,1) = 3: 111111, 1113, 15.
%e A307432 T(6,2) = 1: 222.
%e A307432 T(6,3) = 1: 33.
%e A307432 T(6,6) = 1: 6.
%e A307432 Triangle T(n,k) begins:
%e A307432    1;
%e A307432    0, 1;
%e A307432    0, 1, 1;
%e A307432    1, 1, 0, 1;
%e A307432    1, 2, 1, 0, 1;
%e A307432    3, 2, 1, 0, 0, 1;
%e A307432    5, 3, 1, 1, 0, 0, 1;
%e A307432    9, 4, 1, 0, 0, 0, 0, 1;
%e A307432   11, 6, 3, 0, 1, 0, 0, 0, 1;
%e A307432   18, 6, 3, 1, 1, 0, 0, 0, 0, 1;
%e A307432   27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1;
%e A307432   ...
%p A307432 b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
%p A307432       b(n, i-1, k)+b(n-i, min(n-i, i), Bits[And](i, k))))
%p A307432     end:
%p A307432 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(
%p A307432          b(n$2, `if`(n=0, 0, 2^ilog2(2*n)-1))):
%p A307432 seq(T(n), n=0..14);
%Y A307432 Column k=0 gives A307435.
%Y A307432 Row sums give A000041.
%Y A307432 Main diagonal gives A000012.
%Y A307432 Cf. A050314 (the same for XOR), A307431 (the same for OR).
%K A307432 nonn,tabl,look,base
%O A307432 0,12
%A A307432 _Alois P. Heinz_, Apr 08 2019