This is an exercise of Enumerative Combinatorics Chapter $2$
I know I have to do inclusion-exclusion for the upper and lower bound, but I have no idea how to solve it
Fix $j,k\ge 1$ for $n\ge 0$. let $f(n)$ be the number of integer sequences $\{a_n\}$ such that $1\leq a_i \leq k \text{ for }1\leq i \leq n \text{ and }a_i\geq a_{i-1}-j $ for $2 \leq i \leq n$ proof that $$ F(x):=\sum{f(n)x^n}=\sum_{n\geq 0}\frac{1}{\sum_{i\geq 0}(-1)^i\binom{k-j(i-1)}{i}x^i} $$ Could you help me or give some hints?