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.