Help me please to find a general coefficient $a_j$ of the following series $$ \left(\sum_{j=0}^{\infty}\frac{1}{j!}\left(\frac{t^2}{8p}\right)^j\right)\left(\sum_{j=0}^k\frac{(-1)^jt^{2j}}{4^jp^{2j}j!(n+j)!}\right)=1+\sum_{j}\frac{a_jt^{2j}}{p^j}. $$ Here $n \in N, p\geq2, k\in N$.
Thank you for your help.
The coefficient of $t^{2m}$ in $\displaystyle\sum_{i=0}^\infty \dfrac{t^{2i}}{i!(8p)^i} \sum_{j=0}^\infty \dfrac{(-1)^j t^{2j}}{4^j p^{2j} j!(n+j)!}$ is, according to Maple, $\dfrac{\text{LaguerreL}(m,n,2/p)}{(8p)^m (m+n)!}$, where $\text{LaguerreL}(m,n,t)$ is a generalized Laguerre polynomial.