Euler numbers
In mathematics, the Euler numbers are a sequence En of integers (sequence in the OEIS) defined by the 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 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. This article 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.
An explicit formula for Euler numbers is:
where i denotes the imaginary unit with i2 = −1.
The Euler number E2n can be expressed as a sum over the even partitions of 2n,
as well as a sum over the odd partitions of 2n − 1,
where in both cases K = k1 + ··· + kn and
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