A177974 Triangle T(n,k) = n!/(n - k)! + n!/k! - n! read by rows, 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 0, 4, 1, 1, 5, -40, -40, 5, 1, 1, 6, -330, -480, -330, 6, 1, 1, 7, -2478, -3990, -3990, -2478, 7, 1, 1, 8, -20104, -33264, -36960, -33264, -20104, 8, 1, 1, 9, -181368, -301896, -344736, -344736, -301896, -181368, 9, 1

Offset

0,5

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

T(n,k) = n!/(n - k)! + n!/k! - n.

T(n,k) = A155162(n,k) - n!. - Seiichi Manyama, Sep 26 2023

Example

Triangle begins:
  1;
  1, 1;
  1, 2,       1;
  1, 3,       3,       1;
  1, 4,       0,       4,       1;
  1, 5,     -40,     -40,       5,       1;
  1, 6,    -330,    -480,    -330,       6,       1;
  1, 7,   -2478,   -3990,   -3990,   -2478,       7,       1;
  1, 8,  -20104,  -33264,  -36960,  -33264,  -20104,       8, 1;
  1, 9, -181368, -301896, -344736, -344736, -301896, -181368, 9, 1;
  ...

Mathematica

Flatten[Table[Table[n!/(n - k)! + n!/k! - n!, {k, 0, n}], {n, 0, 10}]]

Crossrefs

Cf. A155162.

Sequence in context: A360335 A370399 A095141 * A095140 A225043 A125605

Adjacent sequences: A177971 A177972 A177973 * A177975 A177976 A177977

Keyword

sign,tabl

Author

Roger L. Bagula, May 16 2010

Status

approved