1. Bernoulli numbers can be expressed through the Euler numbers and vice versa. 2. The coefficients are the Euler numbers of odd and even index, respectively. 3. These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers . 4. The in the expansion of are Euler numbers . 5. The name Euler numbers in particular is sometimes used for a closely related sequence. 6. The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. 7. This asymptotic equation reveals that lies in the common root of both the Bernoulli and the Euler numbers . 8. The errors can in fact be predicted; they are generated by the Euler numbers according to the asymptotic formula 9. See and . ( ) / ( ) are the second ( fractional ) Euler numbers and an autosequence of the second kind. 10. These enumerate the number of alternating permutations on " n " letters and are related to the Euler numbers and the Bernoulli numbers.
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