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Encyclopedia > Euler numbers

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 (sequence A000364 in 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 encyclopedia adheres to the convention adopted above.

The Euler numbers appear in the Taylor series expansion of the secant trigonometric function and the hyperbolic secant (which is the function in the definition), and they also occur in combinatorics.

The Euler polynomials are constructed with the Euler numbers. Results from FactBites:

 Euler Number (1032 words) The Euler Number of a surface is an integer with this properties: Therefore, in order to prove that the results of computing the Euler Number for different divisions yield equal results, it is suffice to prove this for the case in which one division is a refinement of the other. The fact that the Euler Numbers of the Torus and the Sphere are not equal shows that these objects are not topologically equivalent.
More results at FactBites »

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