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