Hellenica World

Euler number

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

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

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; see alternating permutation.

Explicit formulas

An explicit formula for Euler numbers is given by[1]:

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

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 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

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

is a multinomial coefficient. The Kronecker delta's 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,

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

E2n 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

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