Quotients of sequences under the binomial convolution

This paper gives expressions for the solution $\{a(n)\}$ of the equation \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*} where $b(0)\ne 0$, that is, of the equation $a \circ b = c$ in $a$, where $\circ$ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.