%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