I want to solve the following problem:
Determine the coefficients $$[x^n] \frac{1}{1-\sin(\pi x)}$$ up to a multiplicative error of $1+ O(n^{-1})$. In other words, find a function $f(n)$ so that $$[x^n]\frac{1}{1-\sin(\pi x)} = f(n)(1 + O(n^{-1}).$$
From my lecture notes I have the following transfer theorem:
Let $\alpha \in \mathbb{C} \setminus \mathbb{Z_{\leq 0}}$ and assume $f(z)$ is $\Delta$-analytic at 1. Then, if $f(z) = (1-z)^{-\alpha} + o((1-z)^{-\alpha})$ $$ [z^n] f(z) = \frac{n^{\alpha -1}}{\Gamma(\alpha)} ( 1+ O(\frac{1}{n})) + o (n^{\alpha - 1})$$
Now to get $\frac{1}{1- \sin(\pi x)}$ in a form to apply the above theorem, I tried to use Taylor expansion around $x= \frac{1}{2}$ but I do not get the wanted terms.
Can I apply the above theorem for this problem or do I have to solve this problem differently?