I believe I have found an alternative method for compacting the Maclaurin (easily extended to Taylor) series formula that may be preferable to some people.
To find the $n$th-order expansion of $f$, for each combination with replacement of $0$ to $n$ of $f$'s inputs, add a term consisting of that partial of $f$, times the iterated antiderivative of the corresponding differentials. For example, the second-order expansion of $f(x,\ y)$ would be $f + f_x \int dx + f_y \int dy + f_{xx} \int\int dx\ dx + f_{xy} \int\int dx\ dy + f_{yy} \int\int dy\ dy$ (the first term doesn't appear to technically fit the pattern). Note that this method considers combinations, not permutations, so the mixed partial only appears once.
Does this method work in general for any order expansion and any number of inputs $f$ may have?