Euler number

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

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

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 E2n 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

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

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

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.

Generalized Euler numbers

Generalizations of Euler numbers include poly-Euler numbers, which play an important role in multiple zeta functions.
See also

Bell number
Bernoulli number
Euler–Mascheroni constant

References

Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series"[dead link]
Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers 8 (1): A1.

Malenfant, J. "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585.

External links

Hazewinkel, Michiel, ed. (2001), "Euler numbers", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Euler number", MathWorld.

Mathematics Encyclopedia