Question 115
Let $k \ge 3$ be an integer and let functions $f_0, f_1, \dots, f_k$ be defined for positive integers such that
$$
\begin{aligned}
f_j(1) &= 1 && \text{for } 0 \le j \le k, \\
f_0(n+1) &= \sum_{i=0}^{k} f_i(n) && \text{for } n \ge 1 \text{ and} \\
f_{j+1}(n+1) &= f_j(n+1) + f_j(n) && \text{for } 0 \le j \lt k \text{ and } n \ge 1.
\end{aligned}
$$
Find integers $a_0, a_1, \dots, a_k$ such that for all $n \ge 1$
$$
f_k(n + k + 1) = \sum_{i=0}^{k} a_i f_k(n + i).
$$
$$
\begin{aligned}
f_j(1) &= 1 && \text{for } 0 \le j \le k, \\
f_0(n+1) &= \sum_{i=0}^{k} f_i(n) && \text{for } n \ge 1 \text{ and} \\
f_{j+1}(n+1) &= f_j(n+1) + f_j(n) && \text{for } 0 \le j \lt k \text{ and } n \ge 1.
\end{aligned}
$$
Find integers $a_0, a_1, \dots, a_k$ such that for all $n \ge 1$
$$
f_k(n + k + 1) = \sum_{i=0}^{k} a_i f_k(n + i).
$$