\begin{definition}[Definition:Euler Numbers/Alternative Form/Sequence]
The sequence of the alternative form of Euler numbers begins:
{{begin-eqn}}
{{eqn | l = E_1^*
      | r = 1
      | rr= = -E_2
      | c = 
}}
{{eqn | l = E_2^*
      | r = 5
      | rr= = E_4
      | c = 
}}
{{eqn | l = E_3^*
      | r = 61
      | rr= = -E_6
      | c = 
}}
{{eqn | l = E_4^*
      | r = 1385
      | rr= = E_8
      | c = 
}}
{{eqn | l = E_5^*
      | r = 50 \, 521
      | rr= = -E_{10}
      | c = 
}}
{{eqn | l = E_6^*
      | r = 2 \, 702 \, 765
      | rr= = E_{12}
      | c = 
}}
{{eqn | l = E_7^*
      | r = 199 \, 360 \, 981
      | rr= = -E_{14}
      | c = 
}}
{{eqn | l = E_8^*
      | r = 19 \, 391 \, 512 \, 145
      | rr= = E_{16}
      | c = 
}}
{{eqn | l = E_9^*
      | r = 2 \, 404 \, 879 \, 675 \, 441
      | rr= = -E_{18}
      | c = 
}}
{{eqn | l = E_{10}^*
      | r = 370 \, 371 \, 188 \, 237 \, 525
      | rr= = E_{20}
      | c = 
}}
{{eqn | l = E_{11}^*
      | r = 69 \, 348 \, 874 \, 393 \, 137 \, 901
      | rr= = -E_{22}
      | c = 
}}
{{eqn | l = E_{12}^*
      | r = 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905
      | rr= = E_{24}
      | c = 
}}
{{end-eqn}}
where $E_2, E_4, \ldots$ are the standard form Euler numbers.
{{OEIS|A000364}}
\end{definition}