The **Euler numbers** are a sequence *E _{n}* of integers defined by the following Taylor series expansion:

The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are:

*E*_{0}= 1*E*_{2}= -1*E*_{4}= 5*E*_{6}= -61*E*_{8}= 1,385*E*_{10}= -50,521*E*_{12}= 2,702,765*E*_{14}= -199,360,981*E*_{16}= 19,391,512,145*E*_{18}= -2,404,879,675,441

The Euler numbers appear in the Taylor series expansion of the secant trigonometric function, and they also occur in combinatorics.