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Let $c(n,k)$ denote the number of Dyck paths of semilength $n$ which are contained in the strip $0 \leq y \leq 2k + 1.$ They satisfy the recursion $\sum_{j=0}^{k+1}(-1)^j \binom{2k+2-j}{j}c(n-j,k)=0$ ...

EDIT: This is a case of being too wrapped up in a formulation ($e_j,p_i,$ and the like) to try something simple. It did not occur to me to pull exp to the outside in the weeks I have stared at this. ...