Calculus to Combinatorics

$\begin{aligned} \frac{1}{\pi} \int_0^\frac{\pi}{2} \left(\frac{\sin (2 \theta)}{\cos \theta}\right)^{2016} d\theta & = \frac{1}{\pi} \int_0^\frac{\pi}{2} 2^{2016} \sin ^{2016} \theta \space d\theta \\ & = \frac{2^{2015}}{\pi} \int_0^\frac{\pi}{2} 2 \sin ^{2016} \theta \cos ^{0} \theta \space d\theta \\ & = \frac{2^{2015}}{\pi} B \left(1008\frac{1}{2}, \frac{1}{2} \right) \quad \quad \small \color{#3D99F6}{B (p,q) \text{ is Beta function}} \\ & = \frac{2^{2015} \Gamma \left(1008\frac{1}{2} \right) \Gamma \left(\frac{1}{2} \right)}{\pi \Gamma(1009)}\quad \quad \small \color{#3D99F6}{\Gamma (n) \text{ is Gamma function}} \\ & = \frac{2^{2015} 2016! \sqrt{\pi} \dot{} \sqrt{\pi}}{2^{2016} 1008! \dot{} 1008! \pi} \\ & = \frac{2015!}{1007! 1008!} \\ & = \begin{pmatrix} 2015 \\ 1007 \end{pmatrix} \end{aligned}$

$\Rightarrow 2n+2 = 2\times 1007 + 2 = \boxed{2016}$