The function is a product of matrices ,like:
F(k) = [k, k+1...n-2,n-1,0,....k-1] * A[n]
F(k+1) = [k+1,k+2...n-1,0, 1,.....k] * A[n]
F(k+1) - F(k) = [1,1,1...-n+1,1,1,1] * A[n] = sum(A[n]) - n * A[n - 1 - k];
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