A143200 - OEIS

%I #14 Jun 12 2024 06:54:56

%S 1,1,1,1,0,1,1,-1,-1,1,1,0,0,0,1,1,-1,0,0,-1,1,1,0,-1,0,-1,0,1,1,-1,

%T -1,-1,-1,-1,-1,1,1,0,0,0,0,0,0,0,1,1,-1,0,0,0,0,0,0,-1,1,1,0,-1,0,0,

%U 0,0,0,-1,0,1,1,-1,-1,-1,0,0,0,0,-1,-1,-1,1,1,0,0,0,-1,0,0,0,-1,0,0,0,1

%N Triangle read by rows: T(n, k) = 1 if k = 0 or k = n, T(n, k) = -1 if ( binomial(n, k) mod 2 ) = 1, otherwise T(n, k) = 0.

%C Similar to A047999 but with internal 1's replaced by -1's.

%H G. C. Greubel, <a href="/A143200/b143200.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = -1 if ( binomial(n, k) mod 2 ) = 1, T(n, k) = 1 if k = 0 or k = n, otherwise T(n, k) = 0.

%F Sum_{k=0..n} T(n, k) = A142242(n) (row sums).

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  0,  1;

%e   1, -1, -1,  1;

%e   1,  0,  0,  0,  1;

%e   1, -1,  0,  0, -1,  1;

%e   1,  0, -1,  0, -1,  0,  1;

%e   1, -1, -1, -1, -1, -1, -1,  1;

%e   1,  0,  0,  0,  0,  0,  0,  0,  1;

%e   1, -1,  0,  0,  0,  0,  0,  0, -1,  1;

%e   1,  0, -1,  0,  0,  0,  0,  0, -1,  0,  1;

%t T[n_, k_]:= If[k==0 || k==n, 1, If[Mod[Binomial[n,k], 2]==1, -1, 0]];

%t Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten

%o (Magma)

%o function A143200(n,k)

%o     if k eq 0 or k eq n then return 1;

%o     elif (Binomial(n,k) mod 2) eq 1 then return -1;

%o     else return 0;

%o     end if; end function;

%o [A143200(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 12 2024

%o (SageMath)

%o def A143200(n,k):

%o     if (k==0 or k==n): return 1

%o     elif (binomial(n,k)%2==1): return -1

%o     else: return 0

%o flatten([[A143200(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 12 2024

%Y Cf. A047999, A142242 (row sums).

%K tabl,sign,less

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 20 2008

%E Edited by _N. J. A. Sloane_, Aug 15 2009

%E Edited by _G. C. Greubel_, Jun 12 2024