用数学归纳法证明\(a_{2n+1}=\frac{\alpha ^n}{(2n+1)!}\)

青青子衿

\[\begin{array}{l}
{2^n}{a_n} = 2\left( {n + 1} \right){a_{n + 1}}{a_0} + \sum\limits_{k = 1}^n {2k{a_k}{a_{n + 1 - k}}} \\
{a_{2n + 1}} = \frac{{{\alpha ^n}}}{{\left( {2n + 1} \right)!}}
\end{array}\]