Euler Number
In number theory, the Euler numbers are a sequence En 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.
Read more about Euler Number: Asymptotic Approximation
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