Main Page | See live article | Alphabetical index The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Euler's number, as is the Euler characteristic.) The odd-indexed Euler numbers are all zero. The even-indexed ones 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 encyclopedia adheres to the convention adopted above. The Euler numbers appear in the Taylor series expansion of the secant trigonometric function, and they also occur in combinatorics. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.