Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers

Theorem
Let $m \in \Z_{>0}$ be a positive integer.

Let $\sequence {a_n}$ be the sequence defined as:
 * $a_n = \begin{cases}

0 & : n = 0 \\ 1 & : n = 1 \\ a_{n - 2} + a_{n - 1} + \dbinom {n - 2} m & : n > 1 \end{cases}$

where $\dbinom {n - 2} m$ denotes a binonial coefficient.

Then $\sequence {a_n}$ can be expressed in Fibonacci numbers as:
 * $a_n = F_{m + 1} F_{n - 1} + \paren {F_{m + 2} + 1} F_n - \ds \sum_{k \mathop = 0}^m \dbinom {n + m - k} k$