Euler numbers

In mathematics, the Euler numbers are a sequence E_n of integers defined by the Taylor series expansion
where is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely
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.

Examples

The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are:
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive. This article adheres to the convention adopted above.

Explicit formulas

In terms of Stirling numbers of the second kind

The following two formulas express the Euler numbers in terms of Stirling numbers of the second kind:
where denotes the Stirling numbers of the second kind, and denotes the rising factorial.

As a recursion

The Euler numbers can be defined by the recursion
or equivalently
Both of these recursions can be found by using the fact that

As a double sum

The following two formulas express the Euler numbers as double sums

As an iterated sum

An explicit formula for Euler numbers is
where denotes the imaginary unit with.

As a sum over partitions

The Euler number can be expressed as a sum over the even partitions of,
as well as a sum over the odd partitions of,
where in both cases and
is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the s to and to, respectively.
As an example,

As a determinant

is given by the determinant

As an integral

is also given by the following integrals:

Congruences

W. Zhang obtained the following combinational identities concerning the Euler numbers. For any prime, we have
W. Zhang and Z. Xu proved that, for any prime and integer, we have
where is the Euler's totient function.

Lower bound

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

Euler zigzag numbers

The Taylor series of is
where is the Euler zigzag numbers, beginning with
For all even,
where is the Euler number, and for all odd,
where is the Bernoulli number.
For every n,