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A360189 Triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows: T(n,k) = number of nonnegative integers <= n having binary weight k.

%I A360189 #74 Dec 12 2024 23:21:03
%S A360189 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 A360189 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 A360189 1,1,5,8,5,1,1,5,9,5,1,1,5,9,6,1,1,5,9,7,1
%N A360189 Triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows: T(n,k) = number of nonnegative integers <= n having binary weight k.
%C A360189 T(n,k) is defined for all n >= 0 and k >= 0.  Terms that are not in the triangle are zero.
%H A360189 Alois P. Heinz, <a href="/A360189/b360189.txt">Rows n = 0..2^12-1, flattened</a>
%H A360189 Peter J. Taylor, <a href="https://mathoverflow.net/a/483260/231922">Closed form for Sum_{k=0..n} [wt(k) = m] where wt(n) is the binary weight of n</a>, answer to question on MathOverflow (2024).
%H A360189 Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%F A360189 T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.
%F A360189 T(2^n-1,k) = A007318(n,k) = binomial(n,k).
%F A360189 T(n,floor(log_2(n+1))) = A090996(n+1).
%F A360189 Sum_{k>=0} T(n,k) = n+1.
%F A360189 Sum_{k>=0} k * T(n,k) = A000788(n).
%F A360189 Sum_{k>=0} k^2 * T(n,k) = A231500(n).
%F A360189 Sum_{k>=0} k^3 * T(n,k) = A231501(n).
%F A360189 Sum_{k>=0} k^4 * T(n,k) = A231502(n).
%F A360189 Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).
%F A360189 Sum_{k>=0} 3^k * T(n,k) = A130665(n).
%F A360189 Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).
%F A360189 Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).
%F A360189 Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).
%F A360189 Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).
%F A360189 Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).
%F A360189 Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).
%F A360189 Sum_{k>=0} n^k * T(n,k) = A361257(n).
%F A360189 T(n,k) = Sum_{j=0..min(k, A000120(n+1)-1)} binomial(A272020(n+1,j+1)-1,k-j) for n >= 0, k >= 0 (see _Peter J. Taylor_ link). - _Mikhail Kurkov_, Nov 27 2024
%e A360189 T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.
%e A360189 T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.
%e A360189 Triangle T(n,k) begins:
%e A360189   1;
%e A360189   1, 1;
%e A360189   1, 2;
%e A360189   1, 2, 1;
%e A360189   1, 3, 1;
%e A360189   1, 3, 2;
%e A360189   1, 3, 3;
%e A360189   1, 3, 3, 1;
%e A360189   1, 4, 3, 1;
%e A360189   1, 4, 4, 1;
%e A360189   1, 4, 5, 1;
%e A360189   1, 4, 5, 2;
%e A360189   1, 4, 6, 2;
%e A360189   1, 4, 6, 3;
%e A360189   1, 4, 6, 4;
%e A360189   1, 4, 6, 4, 1;
%e A360189   ...
%p A360189 b:= proc(n) option remember; `if`(n<0, 0,
%p A360189       b(n-1)+x^add(i, i=Bits[Split](n)))
%p A360189     end:
%p A360189 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A360189 seq(T(n), n=0..23);
%o A360189 (PARI) T(n,k) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(j=0, min(k,#v1-1), binomial(v1[j+1]-1,k-j)) \\ _Mikhail Kurkov_, Nov 27 2024
%Y A360189 Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).
%Y A360189 Last elements of rows give A090996(n+1).
%Y A360189 Cf. A000027, A000120, A000225, A000788, A006046, A007318, A116520, A116522, A116525, A116526, A130665, A130667, A161342, A231500, A231501, A231502, A272020, A361257.
%K A360189 nonn,look,tabf,base
%O A360189 0,5
%A A360189 _Alois P. Heinz_, Mar 04 2023