# Higher-order derivatives of $(e^x + e^{-x})^{-1}$

MathOverflow Asked by tobias on November 28, 2020

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
$$P_{n+1}(x) = P_n'(x) cdot x cdot (x^2+1) – P_n(x)((2cdot n +1)cdot x^2-1).$$
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 $$sum_{i=0}^n |a_{2i}^{(n)}| = n! cdot 2^n$$.

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. The OEIS also has a separate entry for the maximal coefficients 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.

Correct answer by Timothy Chow on November 28, 2020

## Related Questions

### Example of an intersection complex not concentrated in a single degree

1  Asked on November 29, 2021

### Is every proper regular relative algebraic space curve over a Dedekind domain projective?

1  Asked on November 26, 2021 by lisa-s

### Examples of residually-finite groups

7  Asked on November 26, 2021 by yiftach-barnea

### Reference: Packing under translation is in NP

1  Asked on November 26, 2021 by till

### Representing a continuous time-inhomogeneous Markov chain by a stochastic integral

0  Asked on November 26, 2021

### Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps

0  Asked on November 26, 2021 by sid-a

### Values of a pair of determinants

1  Asked on November 26, 2021

### What subsystem of third order arithmetic proves the real numbers are Dedekind complete?

1  Asked on November 26, 2021

### Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

1  Asked on November 24, 2021

### Error rate implying regularity

0  Asked on November 24, 2021 by user69642

### Pairing up vertices in a graph

1  Asked on November 24, 2021

### Minimizing an f-divergence and Jeffrey’s Rule

0  Asked on November 24, 2021 by jw7642

### Rational functions with trivial Weil symbols at every point

0  Asked on November 24, 2021 by daniil-rudenko

### Regular singular point of non-linear ODE: $dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$

1  Asked on November 24, 2021

### Inverse problem of Chern Classes

4  Asked on November 22, 2021 by temitope-a

### Reference for graduate-level text or monograph with focus on “the continuum”

5  Asked on November 22, 2021 by ruth-no

### Do Poincaré residue and integrable log connection commute?

0  Asked on November 22, 2021

### Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,zinmathbb Z$

0  Asked on November 22, 2021

### How to solve a differential equation in the form $frac{partial}{partial t}g(x,t)=g(x-Delta,t)+frac{partial^2}{partial x^2} g(x,t)$?

1  Asked on November 22, 2021

### Multidimensional series: an application of quantum field theory

0  Asked on November 22, 2021 by andrea-t