I'm wondering, does anyone know whether there is an algorithm, or some sort of solver, that takes a function and returns an expression for the coefficient of the nth term of its Taylor series? For example the funciton $e^x$ will give us the expression $1/n!$. This is fairly simple to derive using the definition of the taylor expansions, but other functions are arduous and require manipulation of generalized binomial coefficients and application of other rules. Anyone has an idea of how to mechanize the process? There are integral solvers online, even though integration is known not to be a strictly mechanical process like derivation, so why not for this?
My goal is to find the nth coefficients of a taylor expansion efficiently. If there does not exist a solver to find the expression for the nth coefficient, then I also find satisfactory, as an answer, an algorithm that takes any function and computes its nth derivative at $0$ in $O(n)$ time, which would allow me to find the nth taylor coefficient fairly efficiently.