Euler number

For other uses, see List of things named after Leonhard Euler#Euler's numbers.

In number theory, the Euler numbers are a sequence E_n of integers (sequence A122045 in OEIS) defined by the following Taylor series expansion:

\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty \frac{E_n}{n!} \cdot t^n\!

where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in OEIS) have alternating signs. Some values are:

E₀ = 1

E₂ = −1

E₄ = 5

E₆ = −61

E₈ = 1,385

E₁₀ = −50,521

E₁₂ = 2,702,765

E₁₄ = −199,360,981

E₁₆ = 19,391,512,145

E₁₈ = −2,404,879,675,441

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.

Explicit formulas

Iterated sum

An explicit formula for Euler numbers is:^[1]

E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k {k\choose j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}

where i denotes the imaginary unit with i²=−1.

Sum over partitions

The Euler number E_2n can be expressed as a sum over the even partitions of 2n,^[2]

E_{2n} = (2n)! \sum_{0 \leq k_1, \ldots, k_n \leq n}~ \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \delta_{n,\sum mk_m } \left( \frac{-1~}{2!} \right)^{k_1} \left( \frac{-1~}{4!} \right)^{k_2} \cdots \left( \frac{-1~}{(2n)!} \right)^{k_n} ,

as well as a sum over the odd partitions of 2n − 1,^[3]

E_{2n} = (-1)^{n-1} (2n-1)! \sum_{0 \leq k_1, \ldots, k_n \leq 2n-1} \left( \begin{array} {c} K \\ k_1, \ldots , k_n \end{array} \right) \delta_{2n-1,\sum (2m-1)k_m } \left( \frac{-1~}{1!} \right)^{k_1} \left( \frac{1}{3!} \right)^{k_2} \cdots \left( \frac{(-1)^n}{(2n-1)!} \right)^{k_n} ,

where in both cases $K =k_1 + \cdots + k_n$ and

\left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \equiv \frac{ K!}{k_1! \cdots k_n!}

is a multinomial coefficient. The Kronecker delta's in the above formulas restrict the sums over the k's to $2k_1 + 4k_2 + \cdots +2nk_n=2n$ and to $k_1 + 3k_2 + \cdots +(2n-1)k_n=2n-1$ , respectively.

As an example,

\begin{align} E_{10} & = 10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!} - \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \\ & = 9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!} +\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) \\ & = -50,521. \end{align}

Determinant

E_2n is also given by the determinant

\begin{align} E_{2n} &=(-1)^n (2n)!~ \begin{vmatrix} \frac{1}{2!}& 1 &~& ~&~\\ \frac{1}{4!}& \frac{1}{2!} & 1 &~&~\\ \vdots & ~ & \ddots~~ &\ddots~~ & ~\\ \frac{1}{(2n-2)!}& \frac{1}{(2n-4)!}& ~&\frac{1}{2!} & 1\\ \frac{1}{(2n)!}&\frac{1}{(2n-2)!}& \cdots & \frac{1}{4!} & \frac{1}{2!}\end{vmatrix}. \end{align}

Asymptotic approximation

The Euler numbers grow quite rapidly for large indices as they have the following lower bound

|E_{2 n}| > 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e}\right)^{2 n}.

Euler zigzag numbers

The Taylor series of $\sec x+\tan x$ is $\sum_{n=0}^{\infty} \frac{A_n}{n!}x^n$ , where $A_n$ is the Euler zigzag numbers, beginning with

1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... (sequence A000111 in OEIS)

For all even n, $A_n$ = $(-1)^{\frac{n}{2}}E_n$ , where $E_n$ is the Euler number, and for all odd n, $A_n$ = $(-1)^{\frac{n-1}{2}}\frac{2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}$ , where $B_n$ is the Bernoulli number.