\displaystyle \sum_{n=0}^{\infty}\left [(n+1)(n+2) a_{n+2} - a_n] x^n=\sum_{n=0}^{\infty}\left [(-1)^{n+1}\frac{x^{n+1}}{(2n+1)!}] x^n
\displaystyle \sum_{n=0}^{\infty}\left [(n+1)(n+2) a_{n+2} - a_n] x^n=\sum_{n=0}^{\infty}\left [(-1)^{n+1}\frac{x^{n+1}}{(2n+1)!}] x^n