If $(1-\cos{x})^{n+1}=\sum_{k=0}^\infty c_k\cos{kt}$ for some integers $c_k$, then what is $\sum_{k=0}^\infty kc_k$?

Question

This just pop unto my mind. So maybe you guys can solve this $$(1-\cos{x})^{n+1}=\sum_{k=0}^\infty c_k\cos{kt}$$ for some $c_k\in\mathbb{Z}$. The question is. What is the value of $$\sum_{k=0}^\infty k\cdot c_k$$

Answer 1

Cleary, for $n=0$: $c_0=1=-c_1$ and $c_i=0$ for $i>1$.

Observe that $$(1-\cos(x)) \cos(ax) = \cos(ax)- \frac 1 2 [\cos(1-a)x)+ \cos(1+ax)].$$

Then proceed by induction.