What's this function?

$P=\displaystyle \sum^{n}_{r=0}$ $^{n}C_{r} sin(rx) cos(n-r)x$

$P=\displaystyle \sum^{n}_{r=0}$ $~^{n}C_{n-r} sin(n-r)x cos(r)x$

adding

$\large{2P=(^{n}C_{0}+^{n}C_{1}+^{n}C_{2}+^{n}C_{3}+....+^{n}C_{n})sin(nx)}$

$2P=2^{n}sin(nx)$

$f(n)=2^{n-1}$

$f(15)=2^{14}=16384$

Let $\sum_{k=0}^n \binom{n}{k} \sin(kx) \cos[(n-k)x] = G(x)$ . It is obvious that

$G(x) = 2^{(n-1)} \times \sin(nx) \forall n$ positive odd integer. Therefore $f(15) = 2^{14} = \boxed{16384}$ .