Prove that \( 1^{2} \cdot C_{0}+2^{2} \cdot C_{1}+3^{2} \cdot C_{2}...

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Prove that \( 1^{2} \cdot C_{0}+2^{2} \cdot C_{1}+3^{2} \cdot C_{2}+4^{2} \cdot C_{3}+\ldots .+(n+1)^{2} C_{n}=2^{n-2}(n+1)(n+4) \).
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