In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 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 A028296 in the OEIS) have alternating signs. Some values 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
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/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.
Sum over partitions
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 and
is a multinomial coefficient. The Kronecker delta's in the above formulas restrict the sums over the k's to and to , respectively.
As an example,
E2n is also given by the determinant
The Euler numbers grow quite rapidly for large indices as they have the following lower bound
Euler zigzag numbers
The Taylor series of is , where 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 is the Euler number, and for all odd n, = , where is the Bernoulli number.
For every n,
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.
- ^ Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" Archived May 11, 2012, at the Wayback Machine.
- ^ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1.
- ^ Malenfant, J. "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585.
- ^  Hassan Jolany, Roberto B. Corcino, Takao Komatsu, "More properties on multi-poly-Euler polynomials, Boletín de la Sociedad Matemática Mexicana October 2015, Volume 21, Issue 2, pp 149-162"