Quotient of Complicated Products

Algebra Level 3

For each positive integer nn, let an=n22n+1a_n=\dfrac{n^2}{2n+1}. Furthermore, let PP and QQ be real numbers such that P=a1a2a2013,Q=(a1+1)(a2+1)(a2013+1).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 QP\frac QP?

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