Sum of Euler Numbers by Binomial Coefficients Vanishes/Examples

Example of Use of Sum of Euler Numbers by Binomial Coefficients Vanishes

 * $\begin{array}{r|cccccccccc}

E_k & \dbinom n 0 & & \dbinom n 2 & & \dbinom n 4 & & \dbinom n 6 & & \dbinom n 8 & & \dbinom n {10} \\ \hline E_0 = +1        & 1 E_0  &  &  &  &  &  &  &  &  &  &  &  = 1 \\ E_2 = -1        & 1 E_0  & + & 1 E_2 & & &  & &  & &  &  & = 0 \\ E_4 = +5        & 1 E_0  & + & 6 E_2 & + & 1 E_4 &  &  & & & & &  = 0 \\ E_6 = -61       & 1 E_0  & + & 15 E_2 & + & 15 E_4 & + & 1 E_6 &  &  & & & = 0 \\ E_8 = +1385     & 1 E_0  & + & 28 E_2 & + & 70 E_4 & + & 28 E_6 & + & 1 E_8 &  & & = 0 \\ E_{10} = -50521 & 1 E_0  & + & 45 E_2 & + & 210 E_4 & + & 210 E_6 & + & 45 E_8 & + & 1 E_{10} & = 0 \\ \end{array}$