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A360189
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.
15
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, 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, 1, 1, 5, 8, 5, 1, 1, 5, 9, 5, 1, 1, 5, 9, 6, 1, 1, 5, 9, 7, 1
OFFSET
0,5
COMMENTS
T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.
LINKS
Wikipedia, Iverson bracket
FORMULA
T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.
T(2^n-1,k) = A007318(n,k) = binomial(n,k).
T(n,floor(log_2(n+1))) = A090996(n+1).
Sum_{k>=0} T(n,k) = n+1.
Sum_{k>=0} k * T(n,k) = A000788(n).
Sum_{k>=0} k^2 * T(n,k) = A231500(n).
Sum_{k>=0} k^3 * T(n,k) = A231501(n).
Sum_{k>=0} k^4 * T(n,k) = A231502(n).
Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).
Sum_{k>=0} 3^k * T(n,k) = A130665(n).
Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).
Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).
Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).
Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).
Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).
Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).
Sum_{k>=0} n^k * T(n,k) = A361257(n).
EXAMPLE
T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.
T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.
Triangle T(n,k) begins:
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, 4, 5, 2;
1, 4, 6, 2;
1, 4, 6, 3;
1, 4, 6, 4;
1, 4, 6, 4, 1;
...
MAPLE
b:= proc(n) option remember; `if`(n<0, 0,
b(n-1)+x^add(i, i=Bits[Split](n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..23);
CROSSREFS
Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).
Last elements of rows give A090996(n+1).
Sequence in context: A249809 A075104 A253667 * A368210 A233932 A008289
KEYWORD
nonn,look,tabf,base
AUTHOR
Alois P. Heinz, Mar 04 2023
STATUS
approved