A360753 Matrix inverse of A360657.

1, 0, 1, 0, -2, 1, 0, 1, -5, 1, 0, 1, 8, -9, 1, 0, 2, 4, 29, -14, 1, 0, 6, 4, -10, 75, -20, 1, 0, 24, 4, -41, -115, 160, -27, 1, 0, 120, -8, -147, -196, -490, 301, -35, 1, 0, 720, -136, -624, -392, -231, -1484, 518, -44, 1

Offset: 0

Author

Werner Schulte, Feb 21 2023

Keywords

• sign
• easy
• tabl

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :  0    1     2     3     4     5      6    7    8  9
=========================================================
  0 :  1
  1 :  0    1
  2 :  0   -2     1
  3 :  0    1    -5     1
  4 :  0    1     8    -9     1
  5 :  0    2     4    29   -14     1
  6 :  0    6     4   -10    75   -20      1
  7 :  0   24     4   -41  -115   160    -27    1
  8 :  0  120    -8  -147  -196  -490    301  -35    1
  9 :  0  720  -136  -624  -392  -231  -1484  518  -44  1
  etc.
		

Crossrefs

Cf. A132013, A215534, A354794, A354795, A360657 (matrix inverse).

Programs

PARI

tabl(m) = {my(n=2*m, A = matid(n), B, C, T); for( i = 2, n, for( j = 2, i, A[i, j] = A[i-1, j-1] + j * A[i-1, j] ) ); B = A^(-1); C = matrix( m, m, i, j, if( j == 1, 0^(i-1), sum( r = 0, i-j, B[i-j+1, r+1] * A[i-1+r, i-1] ) ) ); T = 1/C; }

Formula

Conjectured formulas:

1. Matrix product of A354794 and T without column 0 equals A215534.

2. Matrix product of T and A354794 without column 0 equals A132013.

3. E.g.f. of column k > 0: Sum_{n >= k} T(n, k) * t^(n-1) / (n-1)! = (1 - t) * (Sum_{n >= k} A354795(n, k) * t^(n-1) / (n-1)!).