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.
E0 = 1 E2 = −1 E4 = 5 E6 = −61 E8 = 1385 E10 = −50521 E12 = 2702765 E14 = −199360981 E16 = 19391512145 E18 = −2404879675441
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.
In terms of Stirling numbers of the second kind
Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind
As an iterated sum
where i denotes the imaginary unit with i2 = −1.
As a sum over partitions
The Euler number E2n can be expressed as a sum over the even partitions of 2n,
where in both cases K = k1 + ··· + kn and
As an example,
As a determinant
E2n is given by the determinant
As an integral
E2n is also given by the following integrals:
where is the Euler's totient function.
The Euler numbers grow quite rapidly for large indices as they have the following lower bound
Euler zigzag numbers
The Taylor series of sec x + tan x is
where An is the Euler zigzag numbers, beginning with
- 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... (sequence A000111 in the OEIS)
For all even n,
where En is the Euler number; and for all odd n,
where Bn is the Bernoulli number.
For every n,
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