A360753 - OEIS

%I #23 Feb 24 2023 01:56:42

%S 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,

%T 24,4,-41,-115,160,-27,1,0,120,-8,-147,-196,-490,301,-35,1,0,720,-136,

%U -624,-392,-231,-1484,518,-44,1

%N Matrix inverse of A360657.

%F Conjectured formulas:

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

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

%F 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)!).

%e Triangle T(n, k) for 0 <= k <= n starts:

%e n\k :  0    1     2     3     4     5      6    7    8  9

%e =========================================================

%e   0 :  1

%e   1 :  0    1

%e   2 :  0   -2     1

%e   3 :  0    1    -5     1

%e   4 :  0    1     8    -9     1

%e   5 :  0    2     4    29   -14     1

%e   6 :  0    6     4   -10    75   -20      1

%e   7 :  0   24     4   -41  -115   160    -27    1

%e   8 :  0  120    -8  -147  -196  -490    301  -35    1

%e   9 :  0  720  -136  -624  -392  -231  -1484  518  -44  1

%e   etc.

%o (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; }

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

%K sign,easy,tabl

%O 0,5

%A _Werner Schulte_, Feb 21 2023