How do I remember the coefficients of a Taylor expansion?

If it's not sticking in your head; don't. I find it's easier to just remember what the point of a Taylor expansion is: to express a function as an infinite polynomial, often because polynomials are easy to differentiate. It's this ease of differentiation that is how we can find the coefficients, and the differentiation is easy enough that just remembering "differentiate and see what happens" can be enough to get you the coefficients.

More specifically, we know that our function as a taylor series is f(x) = a + bx + cx^2 + dx^3 + ex^4 + ... . If we wanted a, we'd just set x to 0 to get rid of everything else, and this works because a is constant. But since it's polynomial, we can make any of the terms constant by differentiating the right number of times. Then we can make x = 0 and we have our constant in terms of f(0). For example, if I want the coefficient of x^4, I can just differentiate both sides four times (on the RHS just differentiating the x^4 term since we know everything else will disappear anyway), set x to 0, and do a bit of rearranging. In this case, I'd get f''''(0)=4321e, so e=f''''(0)/4!. Indeed, for the coefficient of x^n, this argument gives us c_n=f^(n)(0)/n!. You might want to try proving this by induction to fully get why this is.