In the area of number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, the Euler numbers are a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 E_n of integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s defined by the following Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 expansion:


where cosh t is the hyperbolic cosine

Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero

0 (number)

0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

. The even-indexed ones 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

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 expansions of the secant

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

; see alternating permutation.

Explicit formulas

An explicit formula for Euler numbers is given by:

The Euler number E_2n can be expressed as a sum over the even partitions

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

 of 2n,

as well as as a sum over the odd partitions of 2n-1,
where in both cases and
is a multinomial coefficient. The Kronecker delta's restrict the sums over the ks to and to
, respectively. As an example,

E_2n is also given by the determinant

Asymptotic approximation

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