The Lehmer-Comtet sequence The Lehmer-Comtet sequence is a sequence such that a(n) is the nth derivative of f(x) = xx with respect to x as evaluated at x = 1.

Task

Take a non-negative integer as input and output the nth term of the  Lehmer-Comtet sequence.

This is code-golf so you should minimize the file size of your source code.

Test Cases

OEIS 5727

Here are the first couple terms in order (copied from the OEIS)

1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880
Haskell, 77 75 bytes, no differentiation builtins

x@(a:b)&amp;y@(c:d)=a*c:zipWith(+)(b&amp;y)(x&amp;d)
s=1:s&amp;(1:scanl(*)1[-1,-2..])
(s!!)

Try it online!
How it works
We represent a function as its infinite list of Taylor series coefficients about \$x = 1\$:
$$
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(1)}{n!} (x - 1)^n
$$
is represented by \$[f(1), f'(1), f''(1), ...]\$.
The &amp; operator multiplies two such functions using the product rule.  This lets us recursively define the function \$s(x) = x^x\$ in terms of itself using the differential equation
$$
s(1) = 1, \\
s'(x) = s(x) \cdot (1 + \ln x),
$$
where
$$
\ln x = \sum_{n=1}^\infty \frac{(-1)^{n-1}(n - 1)!}{n!}(x - 1)^n.
$$