A085143 Triangle table from number wall of reversion of Fibonacci numbers.

1, -1, -1, -1, -1, -1, 1, 0, 2, 1, 1, -2, 4, 3, 1, -1, 3, -11, -5, -5, -1, -1, -1, -34, 10, -20, -8, -1, 1, 11, 106, -116, 96, 44, 13, 1, 1, 15, 368, -328, 716, 86, 125, 21, 1, -1, 13, -1324, -1344, -5634, 1866, -1063, -316, -34, -1, -1, 77, -4811, -17235

Offset: 1

Author

Michael Somos, Jun 19 2003

Keywords

• sign
• tabl

Examples

			T(4,2)=0 since det([0,0,1,-1; 0,1,-1,0; 1,-1,0,2; -1,0,2,-3])=0.
    1
   -1    -1
   -1    -1    -1
    1     0     2     1
    1    -2     4     3     1
   -1     3   -11    -5    -5    -1
   -1    -1   -34    10   -20    -8    -1
    1    11   106  -116    96    44    13     1
    1    15   368  -328   716    86   125    21     1
   -1    13 -1324 -1344 -5634  1866 -1063  -316   -34    -1
		

Crossrefs

Cf. A007440.

Programs

Maple

A085143 := proc(n,k)
    local A,r,c ;
    A := Matrix(n,n) ;
    for r from 1 to n do
    for c from 1 to n do
        A[r,c] :=  A007440(r+c-1+k-n) ;
    end do:
    end do:
    Determinant(A) ;
end proc:
seq(seq(A085143(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Jul 21 2023

PARI

{f(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x),n)} \\ A007440
{T(n,k)=matdet(matrix(n,n,i,j,f(i+j-1+k-n)))}

Formula

T(n, k) = det(f(i+j-1+k-n)_{i, j=1..n}) where f(n)=A007440(n).

T(n, k) = (-1)^[(n+k-1)/2]*T(k-1, n-1) if 1<=k<=n.