A157283 Triangle read by rows: T(n,k) is the determinant of the k X k matrix, values taken from A126988 with columns starting on the first column and rows finishing on the n-th row.

1, 1, 1, 1, 2, 1, 1, 3, -3, 1, 1, 4, 6, 0, 1, 1, 5, -10, -10, -5, 1, 1, 6, 15, -20, 15, -6, 1, 1, 7, -21, 0, 0, -21, -7, 1, 1, 8, 28, 56, 0, 0, 0, 0, 1, 1, 9, -36, 84, 126, 126, 0, 0, 0, 1, 1, 10, 45, 0, -210, 252, -210, 0, 0, -10, 1, 1, 11, -55, -165, 330, 0, 0, 330, 165, -55, -11, 1

Offset

0,5

Comments

The absolute values of the nonzero values appear to be as in Pascal's triangle A007318. The second diagonal appears to be zero at nonsquare free column indices. There are a lot of zeros in the middle as the table gets bigger.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Example

Table begins:
  1
  1    1
  1    2    1
  1    3   -3    1
  1    4    6    0    1
  1    5  -10  -10   -5    1
  1    6   15  -20   15   -6    1
  1    7  -21    0    0  -21   -7    1
  1    8   28   56    0    0    0    0    1
  1    9  -36   84  126  126    0    0    0    1
  1   10   45    0 -210  252 -210    0    0  -10    1
  ...
T(4,2) = 6 is the determinant of the 2 X 2 matrix:
   [3, 0]
   [4, 2]
T(4,3) = 0 is the determinant of the 3 X 3 matrix:
   [2, 1, 0]
   [3, 0, 1]
   [4, 2, 0]

Maple

T:= (n, k)-> LinearAlgebra[Determinant](Matrix(k,
    (i, j)-> `if`(irem(n-k+i, j, 'q')=0, q, 0))):
seq(seq(T(n, k), k=0..n), n=0..11);  # Alois P. Heinz, Feb 10 2026

Prog

(PARI) T(n, k) = matdet(matrix(k, k, i, j, my(m=n-k+i); if(m%j==0, m/j))) \\ Andrew Howroyd, Feb 10 2026

Crossrefs

Cf. A007318, A126988.

Sequence in context: A183342 A046688 A208342 * A067049 A349976 A090641

Adjacent sequences: A157280 A157281 A157282 * A157284 A157285 A157286

Keyword

sign,tabl

Author

Mats Granvik and Gary W. Adamson, Feb 26 2009

Extensions

Offset and definition corrected by Andrew Howroyd, Feb 10 2026

Status

approved