In mathematics, in the area of number theory, the Euler numbers are a sequence E_n of integers defined by the following Taylor series expansion: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... For other senses of this word, see sequence (disambiguation). ... The integers are commonly denoted by the above symbol. ... As the degree of the Taylor series rises, it approaches the correct function. ...

where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials. In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. ...

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A000364 in OEIS) have alternating signs. Some values are: For other senses of this word, see zero or 0. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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; see alternating permutation. As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci âˆ’ 1 and ci + 1 for any value of i. ...