Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient

Theorem
Let $F_n$ denote the $n$th Fibonacci number.

Then:

where $\dbinom n k$ denotes a binomial coefficient.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $F_{2 n} = \ds \sum_{k \mathop = 1}^n \dbinom n k F_k$

$\map P 0$ is the case:

Thus $\map P 0$ is seen to hold.

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P m$ is true, where $m \ge 1$, then it logically follows that $\map P {m + 1}$ is true.

So this is the induction hypothesis:
 * $F_{2 m} = \ds \sum_{k \mathop = 1}^m \dbinom m k F_k$

from which it is to be shown that:
 * $F_{2 \paren {m + 1} } = \ds \sum_{k \mathop = 1}^{m + 1} \dbinom {m + 1} k F_k$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 0}: F_{2 n} = \ds \sum_{k \mathop = 1}^n \dbinom n k F_k$