In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where \cosh (t) 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 S(n,\ell) denotes the Stirling numbers of the second kind, and denotes the rising factorial.

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 i denotes the imaginary unit with .

As a sum over partitions

The Euler number E2n can be expressed as a sum over the even partitions of 2n , as well as a sum over the odd partitions of 2n − 1 , where in both cases and is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k1 + 4k2 + ··· + 2nkn = 2n and to k1 + 3k2 + ··· + (2n − 1)kn = 2n − 1 , respectively. As an example,

As a determinant

E2n is given by the determinant

As an integral

E2n is also given by the following integrals:

Congruences

W. Zhang obtained the following combinational identities concerning the Euler numbers. For any prime p, we have W. Zhang and Z. Xu proved that, for any prime and integer, we have where \phi(n) 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 An is the Euler zigzag numbers, beginning with For all even n, where En is the Euler number, and for all odd n, where Bn is the Bernoulli number. For every n,