I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$
It is fairly straightforward to obtain
$$ \frac{d^n f}{dx^n} = \frac{P\_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $$
where $P\_n(x)$ is given by the recursive relationship $P\_0(x) = 1$ and
$$P\_{n+1}(x) = P\_n'(x) \cdot x \cdot (x^2+1) - P\_n(x)((2\cdot n +1)\cdot x^2-1). $$
If we represent $P\_n(x)$ by $\sum\_{i = 0}^n a^{(n)}\_{2i} x^{2i}$, then we have $a\_0^{(n)} = 1$ for all $n$, $a\_{2k}^{(n)} = 0$ for all $k > n$, and $$a\_{2i}^{(n+1)} = (2i+1) \cdot a\_{2i}^{(n)} - (2(n-i)+3) \cdot a\_{2 \cdot (i-1)}^{(n)}$$ for all $n \geq 0$ and $0 < i \leq n+1$.
So we can obtain with this relationship, that $a\_{2n}^{(n)} = (-1)^{n}$ and that $a\_2^{(n)} = -3^n +k +1$.
However, I am wondering whether we can say something about the maximum of $|P\_n(e^x)/e^{2n}|$ for each $n$ or the maximum of $\max\_{0 \leq i \leq n} |a\_{2i}^{(n)}|$ over each $n$.


**Edit:** It seems that $\sum\_{i=0}^n |a\_{2i}^{(n)}| = n! \cdot 2^n$.



Higher-order derivatives of $(e^x + e^{-x})^{-1}$
Using the tried-and-true method of calculating small examples and plugging them into the OEIS, one finds that the $P\_n(x)$ are, up to sign, known as MacMahon polynomials, and their coefficients are given by [Eulerian numbers of type B](http://oeis.org/A060187). The OEIS also has a separate entry for the [maximal coefficients](http://oeis.org/A154420) that you are asking about, although it doesn't list a formula for the asymptotic growth. But there is a long list of references that will hopefully be helpful.


