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

%I A085143 #9 Jul 21 2023 07:48:48
%S A085143 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,
%T A085143 -20,-8,-1,1,11,106,-116,96,44,13,1,1,15,368,-328,716,86,125,21,1,-1,
%U A085143 13,-1324,-1344,-5634,1866,-1063,-316,-34,-1,-1,77,-4811,-17235
%N A085143 Triangle table from number wall of reversion of Fibonacci numbers.
%F A085143 T(n, k) = det(f(i+j-1+k-n)_{i, j=1..n}) where f(n)=A007440(n).
%F A085143 T(n, k) = (-1)^[(n+k-1)/2]*T(k-1, n-1) if 1<=k<=n.
%e A085143 T(4,2)=0 since det([0,0,1,-1; 0,1,-1,0; 1,-1,0,2; -1,0,2,-3])=0.
%e A085143     1
%e A085143    -1    -1
%e A085143    -1    -1    -1
%e A085143     1     0     2     1
%e A085143     1    -2     4     3     1
%e A085143    -1     3   -11    -5    -5    -1
%e A085143    -1    -1   -34    10   -20    -8    -1
%e A085143     1    11   106  -116    96    44    13     1
%e A085143     1    15   368  -328   716    86   125    21     1
%e A085143    -1    13 -1324 -1344 -5634  1866 -1063  -316   -34    -1
%p A085143 A085143 := proc(n,k)
%p A085143     local A,r,c ;
%p A085143     A := Matrix(n,n) ;
%p A085143     for r from 1 to n do
%p A085143     for c from 1 to n do
%p A085143         A[r,c] :=  A007440(r+c-1+k-n) ;
%p A085143     end do:
%p A085143     end do:
%p A085143     Determinant(A) ;
%p A085143 end proc:
%p A085143 seq(seq(A085143(n,k),k=1..n),n=1..12) ; # _R. J. Mathar_, Jul 21 2023
%o A085143 (PARI) {f(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x),n)} \\ A007440
%o A085143 {T(n,k)=matdet(matrix(n,n,i,j,f(i+j-1+k-n)))}
%Y A085143 Cf. A007440.
%K A085143 sign,tabl
%O A085143 1,9
%A A085143 _Michael Somos_, Jun 19 2003