Let $\omega$ be a real number between $0$ and $1$, and let:
$$\mathbf{c}\left(n\right)=\binom{\omega+n-1}{n}$$
for all positive integers $n$. Is there a closed form for the Dirichlet inverse $\mathbf{c}^{-1}\left(n\right)$ of this function; i.e., for the function $\mathbf{c}^{-1}\left(n\right)$ so that the formal identity: $$\left(\sum_{m=1}^{\infty}\frac{\mathbf{c}\left(m\right)}{m^{s}}\right)\left(\sum_{n=1}^{\infty}\frac{\mathbf{c}^{-1}\left(n\right)}{n^{s}}\right)=1$$ holds. If so, what is it?