A104984 Matrix inverse of triangle A104980.

1, -1, 1, -1, -2, 1, -3, -1, -3, 1, -13, -3, -1, -4, 1, -71, -13, -3, -1, -5, 1, -461, -71, -13, -3, -1, -6, 1, -3447, -461, -71, -13, -3, -1, -7, 1, -29093, -3447, -461, -71, -13, -3, -1, -8, 1, -273343, -29093, -3447, -461, -71, -13, -3, -1, -9, 1, -2829325, -273343, -29093, -3447, -461, -71, -13, -3, -1, -10, 1

Offset: 0

Author

Paul D. Hanna, Apr 10 2005

Keywords

• sign
• tabl

Comments

Inverse matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

Examples

			Triangle begins:
       1;
      -1,     1;
      -1,    -2,    1;
      -3,    -1,   -3,   1;
     -13,    -3,   -1,  -4,   1;
     -71,   -13,   -3,  -1,  -5,  1;
    -461,   -71,  -13,  -3,  -1, -6,  1;
   -3447,  -461,  -71, -13,  -3, -1, -7,  1;
  -29093, -3447, -461, -71, -13, -3, -1, -8, 1; ...
		

Links

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

Crossrefs

Cf. A104980, A104985 (row sums).

Programs

Mathematica

A003319[n_]:= A003319[n]= If[n==0, 0, n! - Sum[j!*A003319[n-j], {j,n-1}]];
T[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)

PARI

T(n,k)=if(n==k,1,if(n==k+1,-n,-(n-k)!-sum(i=1,n-k-1,i!*T(n-k-i,0))));

Sage

@CachedFunction
def T(n,k):
    if (k==n): return 1
    elif (k==n-1): return -n
    else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2021

Formula

T(n, n) = 1, T(n+1, n) = -(n+1) for n >= 0; otherwise T(n, k) = T(n-k, 0) = -A003319(n-k-1) for n > k+1 and k >= 0.

Sum_{k=0..n} T(n, k) = A104985(n). - G. C. Greubel, Jun 07 2021