\begin{definition}[Definition:Euler Numbers/Alternative Form/Definition 1]
The alternative form Euler numbers $E_n^*$ are a sequence of rational numbers defined as:
{{begin-eqn}}
{{eqn | l = \sech x
      | r = 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {E_n^* x^{2 n} } {\paren {2 n}!}
      | c = 
}}
{{eqn | r = 1 - \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} - \frac {E_3^* x^6} {6!} + \cdots
      | c = 
}}
{{end-eqn}}
where $\size x < \dfrac \pi 2 $.
\end{definition}