In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion where
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.
They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
The even-indexed ones (sequence A028296 in the OEIS) have alternating signs.
Some values are: Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (sequence A000364 in the OEIS).
The following two formulas express the Euler numbers as double sums[3] An explicit formula for Euler numbers is:[4] where i denotes the imaginary unit with i2 = −1.
The Euler number E2n can be expressed as a sum over the even partitions of 2n,[5] as well as a sum over the odd partitions of 2n − 1,[6] where in both cases K = k1 + ··· + kn and is a multinomial coefficient.
The Kronecker deltas in the above formulas restrict the sums over the ks to 2k1 + 4k2 + ··· + 2nkn = 2n and to k1 + 3k2 + ··· + (2n − 1)kn = 2n − 1, respectively.
As an example, E2n is given by the determinant E2n is also given by the following integrals: W. Zhang[7] obtained the following combinational identities concerning the Euler numbers.
The Euler numbers grow quite rapidly for large indices, as they have the lower bound The Taylor series of