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SOME PROPERTIES OF A SEQUENCE ANALOGOUS TO EULER NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 12 June 2012, pp. 425-440
- Print publication:
- June 2013
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- Article
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Let
$\{U_n\}$ 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 
$\{U_n\}$.
