Euler Numbers Calculator

Instructions

Enter an integer n between 0 and 2000 inclusive to calculate the nth euler number.

What is a Euler Number?

Euler numbers are a sequence En of integers defined by the Taylor expansion:

$$( \sum_{n=0}^{\infty} {\frac {E_{n}}{n!}} \cdot t^{n} = {\frac {2}{e^{t}+e^{-t}}} = \displaystyle {\frac {1}{\cosh t}})$$

Euler numbers appear in the coeffients of Euler polynomials and the Taylor series expansions of the secant (sec(x) = 1 / cos(x)) and hyperbolic secant (1 / cosh(x)) functions (through which euler numbers are defined). They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements (also known as zig zag numbers).

How do you calculate a Euler number?

All odd En are equal to zero, while even numbers alternate between positive (n = 4 * k) and negative (n = 4 * k + 2) integers.

The nth euler number can be calculated through several methods.