A360189 - OEIS

%I #70 Mar 06 2023 21:50:35

%S 1,1,1,1,2,1,2,1,1,3,1,1,3,2,1,3,3,1,3,3,1,1,4,3,1,1,4,4,1,1,4,5,1,1,

%T 4,5,2,1,4,6,2,1,4,6,3,1,4,6,4,1,4,6,4,1,1,5,6,4,1,1,5,7,4,1,1,5,8,4,

%U 1,1,5,8,5,1,1,5,9,5,1,1,5,9,6,1,1,5,9,7,1

%N Number T(n,k) of nonnegative integers <= n having binary weight k; triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows.

%C T(n,k) is defined for all n >= 0 and k >= 0.  Terms that are not in the triangle are zero.

%H Alois P. Heinz, <a href="/A360189/b360189.txt">Rows n = 0..2^12-1, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>

%F T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.

%F T(2^n-1,k) = A007318(n,k) = binomial(n,k).

%F T(n,floor(log_2(n+1))) = A090996(n+1).

%F Sum_{k>=0} T(n,k) = n+1.

%F Sum_{k>=0} k * T(n,k) = A000788(n).

%F Sum_{k>=0} k^2 * T(n,k) = A231500(n).

%F Sum_{k>=0} k^3 * T(n,k) = A231501(n).

%F Sum_{k>=0} k^4 * T(n,k) = A231502(n).

%F Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).

%F Sum_{k>=0} 3^k * T(n,k) = A130665(n).

%F Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).

%F Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).

%F Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).

%F Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).

%F Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).

%F Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).

%F Sum_{k>=0} n^k * T(n,k) = A361257(n).

%e T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.

%e T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.

%e Triangle T(n,k) begins:

%e   1;

%e   1, 1;

%e   1, 2;

%e   1, 2, 1;

%e   1, 3, 1;

%e   1, 3, 2;

%e   1, 3, 3;

%e   1, 3, 3, 1;

%e   1, 4, 3, 1;

%e   1, 4, 4, 1;

%e   1, 4, 5, 1;

%e   1, 4, 5, 2;

%e   1, 4, 6, 2;

%e   1, 4, 6, 3;

%e   1, 4, 6, 4;

%e   1, 4, 6, 4, 1;

%e   ...

%p b:= proc(n) option remember; `if`(n<0, 0,

%p       b(n-1)+x^add(i, i=Bits[Split](n)))

%p     end:

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

%p seq(T(n), n=0..23);

%Y Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).

%Y Last elements of rows give A090996(n+1).

%Y Cf. A000027, A000120, A000225, A000788, A006046, A007318, A116520, A116522, A116525, A116526, A130665, A130667, A161342, A231500, A231501, A231502, A361257.

%K nonn,look,tabf,base

%O 0,5

%A _Alois P. Heinz_, Mar 04 2023