Euler Number

Euler Number

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

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.