In the area of number theory, the Euler numbers are a sequence En of integers 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:
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.
An explicit formula for Euler numbers is given by:
The Euler numbers grow quite rapidly for large indices as they have the following lower bound
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