Quotient of Complicated Products

Algebra Level 3

For each positive integer \(n\), let \(a_n=\dfrac{n^2}{2n+1}\). Furthermore, let \(P\) and \(Q\) be real numbers such that \[P=a_1a_2\ldots a_{2013},\qquad Q=(a_1+1)(a_2+1)\ldots(a_{2013}+1).\] What is the sum of the digits of \(\frac QP\)?

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