Solve the recurrence relation  using the generating function technique.

To solve the given recurrence relation using generating functions, we first define the generating function \( A(x) \) for the sequence \( \{a_n\} \) as:

\[ A(x) = \sum_{n=0}^{\infty} a_nx^n \]

Then, we multiply both sides of the recurrence relation by \( x^n \) and sum over all \( n \geq 0 \):

\[ \sum_{n=0}^{\infty} a_nx^n = 2\sum_{n=1}^{\infty} a_{n-1}x^n + \sum_{n=1}^{\infty} n(n-1)x^n \]

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