For a given real number $a$ we define the sequence $\{ E_{ n,a}\}$ by $E_{ 0,a}=1$ and $E_{ n,a}=-a\sum_{ k=1}^{ [n/2]} \binom n{ 2k}E_{ n-2k,a}$ $(n\ge 1)$, where $[x]$ is the greatest integer not exceeding $x$. Since $E_{ n,1}=E_n$ is the n-th Euler number, $E_{ n,a}$ can be viewed as a natural generalization of Euler numbers. In this paper we deduce some identities and an inversion formula involving $\{ E_{ n,a}\}$, and establish congruences for $E_{ 2n,a}\mod{ 2^{ { \rm ord}_2n+8}}$, $E_{ 2n,a}\pmod{ 3^{ { \rm ord}_3n+5}}$ and $E_{ 2n,a}\pmod{ 5^{ { \rm ord}_5n+4}}$ provided that $a$ is a nonzero integer, where ${ \rm ord}_pn$ is the least nonnegative integer $\alpha$ such that $p^{ \a}\mid n$ but $p^{ \a+1}\nmid n$.
Categories: math.NT