A142156 Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k > 0.

1, 1, -1, 2, -1, -1, 6, -2, -1, -3, 24, -6, -2, -3, -13, 120, -24, -6, -6, -13, -71, 720, -120, -24, -18, -26, -71, -461, 5040, -720, -120, -72, -78, -142, -461, -3447, 40320, -5040, -720, -360, -312, -426, -922, -3447, -29093, 362880, -40320, -5040, -2160, -1560

Offset: 0

Author

Paul Curtz, Sep 16 2008

Keywords

• sign
• tabl

Examples

			Triangle begins
    1;
    1,   -1;
    2,   -1,  -1;
    6,   -2,  -1,  -3;
   24,   -6,  -2,  -3, -13;
  120,  -24,  -6,  -6, -13, -71;
  720, -120, -24, -18, -26, -71, -461;
		

Programs

Mathematica

(* b = A003319 *) b[0]=0; b[n_] := b[n] = n! - Sum[k! b[n-k], {k, 1, n-1}]; T[n_, 0] := n!; T[n_, k_] := -(n - k)! b[k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2018 *)

Formula

T(n,0) = A000142(n).

T(n,k) = -A141476(n-1,k-1), k > 0.

Sum_{k=0..n} |T(n,k)| = 2*n! = A098558(n).