In mathematics, the Euler numbers are a sequence E_n of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

where ${\displaystyle \cosh(t)}$ 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.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in the OEIS) 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 (sequence A000364 in the OEIS). This article adheres to the convention adopted above.

The following two formulas express the Euler numbers in terms of Stirling numbers of the second kind:

where ${\displaystyle S(n,\ell )}$ denotes the Stirling numbers of the second kind, and ${\displaystyle x^{\overline {\ell {\phantom {.}}}}=(x)(x+1)\cdots (x+\ell -1)}$ denotes the rising factorial.

The Euler numbers can be defined by the recursion