Someone helps me please, how to calculate:

Question

loser66 · Answer

They give me $E_{10} = 50521$ but no matter how hard I tried, I couldn't get that answer.

loser66 · Answer

$$E_{2n}-\left(\begin{matrix}2n\2n-2\end{matrix}\right)E_{2n-2}+\left(\begin{matrix}2n\2n-4\end{matrix}\right)E_{2n-4}+\cdots+(-1)^n\left(\begin{matrix}2n\2\end{matrix}\right)E_{2}+(-1)^n =0$$

loser66 · Answer

$$E_{2n}=\left(\begin{matrix}2n\2n-2\end{matrix}\right)E_{2n-2}-\left(\begin{matrix}2n\2n-4\end{matrix}\right)E_{2n-4}-\cdots-(-1)^n\left(\begin{matrix}2n\2\end{matrix}\right)E_{2}-(-1)^n $$

loser66 · Answer

$E_0 =1$

loser66 · Answer

If n =1
$$E_2= -(-1)^1 =1$$

loser66 · Answer

$E_4= (-1)^2\left(\begin{matrix}4\2\end{matrix}\right)E_2-(-1)^2\E_4=6E_2-1 = 5$

loser66 · Answer

if n=3
$$E_6=\left(\begin{matrix}6\4\end{matrix}\right)E_4-(-1)^3\left(\begin{matrix}2n\2\end{matrix}\right)E_{2}-(-1)^3\E_6 = 15E_4+15E_2+1=91 $$

loser66 · Answer

But they said E6 =61, how?