A143200 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.

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, -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, 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

Offset

0,1

Comments

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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.

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

Example

Triangle begins as:
  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, -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,  0,  0,  0, -1,  0,  1;

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, If[Mod[Binomial[n, k], 2]==1, -1, 0]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten

Prog

(Magma)
function A143200(n, k)
    if k eq 0 or k eq n then return 1;
    elif (Binomial(n, k) mod 2) eq 1 then return -1;
    else return 0;
    end if; end function;
[A143200(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 12 2024
(SageMath)
def A143200(n, k):
    if (k==0 or k==n): return 1
    elif (binomial(n, k)%2==1): return -1
    else: return 0
flatten([[A143200(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 12 2024

Crossrefs

Cf. A047999, A142242 (row sums).

Sequence in context: A077009 A078556 A144093 * A166282 A047999 A323378

Adjacent sequences: A143197 A143198 A143199 * A143201 A143202 A143203

Keyword

tabl,sign,less

Author

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

Extensions

Edited by N. J. A. Sloane, Aug 15 2009

Edited by G. C. Greubel, Jun 12 2024

Status

approved