OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
FORMULA
Triangle read by rows, A000012 * A047999; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1;..]; and A047999 = Sierpinski's gasket.
The operation takes partial sums of Sierpinski's gasket terms, by columns.
From G. C. Greubel, Dec 02 2024: (Start)
T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).
T(n, 0) = A000027(n+1).
T(n, 1) = A004526(n+1).
T(n, 2) = A004524(n+1).
T(2*n, n) = A080100(n).
Sum_{k=0..n} T(n, k) = A006046(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)
EXAMPLE
First few rows of the triangle =
1;
2, 1;
3, 1, 1;
4, 2, 2, 1;
5, 2, 2, 1, 1;
6, 3, 2, 1, 2, 1;
7, 3, 3, 1, 3, 1, 1;
8, 4, 4, 2, 4, 2, 2, 1;
9, 4, 4, 2, 4, 2, 2, 1, 1;
10, 5, 4, 2, 4, 2, 2, 1, 2, 1;
11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;
12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;
13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;
...
MAPLE
MATHEMATICA
A166556[n_, k_]:= Sum[Mod[Binomial[j, k], 2], {j, k, n}];
Table[A166556[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 02 2024 *)
PROG
(Magma)
A166556:= func< n, k | (&+[(Binomial(j, k) mod 2): j in [k..n]]) >;
[A166556(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 02 2024
(Python)
def A166556(n, k): return sum(binomial(j, k)%2 for j in range(k, n+1))
print(flatten([[A166556(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 02 2024
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Oct 17 2009
STATUS
approved