%I #14 Dec 03 2024 08:24:34

%S 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,

%T 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,

%U 6,6,3,4,2,2,1,4,2,2,1

%N Triangle read by rows, A000012 * A047999.

%H G. C. Greubel, <a href="/A166556/b166556.txt">Rows n = 0..100 of the triangle, flattened</a>

%F 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.

%F The operation takes partial sums of Sierpinski's gasket terms, by columns.

%F From _G. C. Greubel_, Dec 02 2024: (Start)

%F T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).

%F T(n, 0) = A000027(n+1).

%F T(n, 1) = A004526(n+1).

%F T(n, 2) = A004524(n+1).

%F T(2*n, n) = A080100(n).

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

%F Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n))*A006046((n+4)/2) + (1-(-1)^n)*A006046((n+3)/2) ).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)

%e First few rows of the triangle =

%e 1;

%e 2, 1;

%e 3, 1, 1;

%e 4, 2, 2, 1;

%e 5, 2, 2, 1, 1;

%e 6, 3, 2, 1, 2, 1;

%e 7, 3, 3, 1, 3, 1, 1;

%e 8, 4, 4, 2, 4, 2, 2, 1;

%e 9, 4, 4, 2, 4, 2, 2, 1, 1;

%e 10, 5, 4, 2, 4, 2, 2, 1, 2, 1;

%e 11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;

%e 12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;

%e 13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;

%e ...

%p A166556 := proc(n,k)

%p local j;

%p add(A047999(j,k),j=k..n) ;

%p end proc: # _R. J. Mathar_, Jul 21 2016

%t A166556[n_, k_]:= Sum[Mod[Binomial[j,k], 2], {j,k,n}];

%t Table[A166556[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 02 2024 *)

%o (Magma)

%o A166556:= func< n,k | (&+[(Binomial(j,k) mod 2): j in [k..n]]) >;

%o [A166556(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 02 2024

%o (Python)

%o def A166556(n,k): return sum(binomial(j,k)%2 for j in range(k,n+1))

%o print(flatten([[A166556(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Dec 02 2024

%Y Cf. A000027, A004524, A004526, A047999, A080100.

%Y Sums include: A006046 (row), A007729 (diagonal).

%K nonn,easy,tabl

%O 0,2

%A _Gary W. Adamson_, Oct 17 2009