2012-6 Matrix modulo p

Let p be a prime number and let n be a positive integer. Let A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n} be a p^n \times p^n matrix. Prove that A^3 \equiv I \pmod p, where I is the p^n \times p^n identity matrix.

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