- 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 encyclopedia 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
as well as a sum over the odd partitions of 2n − 1,
where in both cases and
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
- 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... (sequence A000111 in OEIS)
For all even n, = , where is the Euler number, and for all odd n, = , where is the Bernoulli number.
Generalized Euler numbers
Generalizations of Euler numbers include 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.
- Hazewinkel, Michiel, ed. (2001), "Euler numbers", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Euler number", MathWorld.