How to prove that the determinant of this matrix is n! (by induction)

Question

A_n is a (n x n) - matrix defined by:
a_ij = 1      if       i=j
       -1      if       j = i - 1
        j²     if       i = j + 1
        0      else
Prove by induction that det(A_n) is n!

Answer 1

for n=3 the matrix is $\pmatrix{1 & 2^2 & 0 \\ -1 & 1 & 3^2 \\ 0 & -1 & 1}$

Calculating the det(A3) gives 14 $\neq$ 3!