In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion , where is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: 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. Examples The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are: {| |E0 ||=||align=right| 1 |- |E2 ||=||align=right| −1 |- |E4 ||=||align=right| 5 |- |E6 ||=||align=right| −61 |- |E8 ||=||align=right| |- |E10 ||=||align=right| |- |E12 ||=||align=right| |- |E14 ||=||align=right| |- |E16 ||=||align=right| |- |E18 ||=||align=right| |} Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive . This article adheres to the convention adopted above. Explicit formulas In terms of Stirling numbers of the second kind Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind where denotes the Stirling numbers of the second kind, and denotes the rising factorial. As a double sum Following two formulas express the Euler numbers as double sums As an iterated sum An explicit formula for Euler numbers is: where denotes the imaginary unit with . As a sum over partitions The Euler number can be expressed as a sum over the even partitions of , as well as a sum over the odd partitions of , where in both cases and is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the s to and to , respectively. As an example, As a determinant is given by the determinant As an integral is also given by the following integrals: Congruences W. Zhang obtained the following combinational identities concerning the Euler numbers, for any prime , we have W. Zhang and Z. Xu proved that, for any prime and integer , we have where is the Euler's totient function. Asymptotic approximation The Euler numbers grow quite rapidly for large indices as they have the following lower bound Euler zigzag numbers The Taylor series of is where is the Euler zigzag numbers, beginning with 1, 1, 1, 2, 5, 16, 61, 272,