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Euler number

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In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

Contents

1 cosh ⁡ t = 2 e t + e − t = ∑ n = 0 ∞ E n n ! ⋅ t n ,

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 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.

As an iterated sum

An explicit formula for Euler numbers is:

E 2 n = i ∑ k = 1 2 n + 1 ∑ j = 0 k ( k j ) ( − 1 ) j ( k − 2 j ) 2 n + 1 2 k i k k

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,

E 2 n = ( 2 n ) ! ∑ 0 ≤ k 1 , … , k n ≤ n ( K k 1 , … , k n ) δ n , ∑ m k m ( − 1 2 ! ) k 1 ( − 1 4 ! ) k 2 ⋯ ( − 1 ( 2 n ) ! ) k n ,

as well as a sum over the odd partitions of 2n − 1,

E 2 n = ( − 1 ) n − 1 ( 2 n − 1 ) ! ∑ 0 ≤ k 1 , … , k n ≤ 2 n − 1 ( K k 1 , … , k n ) δ 2 n − 1 , ∑ ( 2 m − 1 ) k m ( − 1 1 ! ) k 1 ( 1 3 ! ) k 2 ⋯ ( ( − 1 ) n ( 2 n − 1 ) ! ) k n ,

where in both cases K = k1 + ··· + kn and

( K k 1 , … , k n ) ≡ K ! k 1 ! ⋯ k n !

is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k1 + 4k2 + ··· + 2nkn = 2n and to k1 + 3k2 + ··· + (2n − 1)kn = 2n − 1, respectively.

As an example,

E 10 = 10 ! ( − 1 10 ! + 2 2 ! 8 ! + 2 4 ! 6 ! − 3 2 ! 2 6 ! − 3 2 ! 4 ! 2 + 4 2 ! 3 4 ! − 1 2 ! 5 ) = 9 ! ( − 1 9 ! + 3 1 ! 2 7 ! + 6 1 ! 3 ! 5 ! + 1 3 ! 3 − 5 1 ! 4 5 ! − 10 1 ! 3 3 ! 2 + 7 1 ! 6 3 ! − 1 1 ! 9 ) = − 50 521.

As a determinant

E2n is also given by the determinant

E 2 n = ( − 1 ) n ( 2 n ) !   | 1 2 ! 1       1 4 ! 1 2 ! 1     ⋮   ⋱     ⋱       1 ( 2 n − 2 ) ! 1 ( 2 n − 4 ) !   1 2 ! 1 1 ( 2 n ) ! 1 ( 2 n − 2 ) ! ⋯ 1 4 ! 1 2 ! | .

Asymptotic approximation

The Euler numbers grow quite rapidly for large indices as they have the following lower bound

| E 2 n | > 8 n π ( 4 n π e ) 2 n .

Euler zigzag numbers

The Taylor series of sec x + tan x is

∑ n = 0 ∞ A n n ! x n ,

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,

A n = ( − 1 ) n 2 E n ,

where En is the Euler number; and for all odd n,

A n = ( − 1 ) n − 1 2 2 n + 1 ( 2 n + 1 − 1 ) B n + 1 n + 1 ,

where Bn is the Bernoulli number.

For every n,

A n − 1 ( n − 1 ) ! sin ⁡ ( n π 2 ) + ∑ m = 0 n − 1 A m m ! ( n − m − 1 ) ! sin ⁡ ( m π 2 ) = 1 ( n − 1 ) ! .

Generalized Euler numbers

Generalizations of Euler numbers include poly-Euler numbers and multi-poly-Euler numbers, which play an important role in multiple zeta functions.

References

Euler number Wikipedia
(Text) CC BY-SA


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