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Euler number

An Euler number E_n is an integer in the power series for the hyperbolic secant

 

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Explanation

The Euler number is the value of the corresponding Euler polynomial, for argument a half (½). The odd numbers are all equal to zero. The successive numbers with an even index have an alternating sign. The first values of the series 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

 

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Example 1

The power series for the hyperbolic secant is

which you can also write as a sum

where E_n stands for the n-th Euler number. So

 

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History The Swiss mathematician Leonhardt Euler (1707 - 1783) described these numbers.

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