Euler Number

The Euler numbers, also called the secant numbers or zig numbers, are defined for by

(1)

(2)

where is the hyperbolic secant and sec is the secant. For , counts the alternating permutations of objects with one fixed starting orientation, such as . Euler numbers are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.

A different sort of Euler number, the Euler number of a finite complex , is defined by

(3)

This Euler number is a topological invariant.

To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes, and prime Euler numbers are termed Euler number primes.

Some values of the (secant) Euler numbers are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

(OEIS A000364).

With , the slightly different convention defined for by

			(16)
			(17)

is frequently used. These are, for example, the Euler numbers computed by the Wolfram Language function EulerE[n]. This definition has the particularly simple series definition

(18)

and is equivalent to

(19)

where is an Euler polynomial.

The number of decimal digits in for , 2, 4, ... are 1, 1, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, ... (OEIS A047893). The number of decimal digits in for , 1, ... are 1, 5, 139, 2372, 33699, ... (OEIS A103235).

The Euler numbers have the asymptotic series

(20)

A more efficient asymptotic series is given by

(21)

(P. Luschny, pers. comm., 2007).

Expanding for a positive integer gives the identity

(22)

where the coefficient is interpreted as (Ely 1882; Fort 1948; Trott 2004, p. 69) and is a tangent number.

For even nonnegative integers and , Stern (1875) showed that

(23)

iff , where is a positive integer. This result had been previously stated by Sylvester in 1861, but without proof.

For integers and , Shanks (1967, p. 689) defines a generalization of the Euler numbers by

(24)

Here is the Dirichlet L-series

(25)

where is the Jacobi symbol. For , the normalization instead gives

(26)

and is times the coefficient of in the series expansion of . A similar representation using holds for , but this specific form does not apply for . The following table gives the first few values of for , 1, ....

	OEIS
1	A000364	1, 1, 5, 61, ...
2	A000281	1, 3, 57, 2763, ...
3	A000436	1, 8, 352, 38528, ...
4	A000490	1, 16, 1280, 249856, ...
5	A000187	2, 30, 3522, 1066590, ...
6	A000192	2, 46, 7970, 3487246, ...
7	A064068	1, 64, 15872, 9493504, ...
8	A064069	2, 96, 29184, 22634496, ...
9	A064070	2, 126, 49410, 48649086, ...
10	A064071	2, 158, 79042, 96448478, ...