SOME PROPERTIES OF A SEQUENCE ANALOGOUS TO EULER NUMBERS

Abstract

AbstractLet $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.