Polynomials of Fibonacci Numbers

The Fibonacci numbers ${f_{n}, n \geq 0}$ are defined recursively as $f_{n} = f_{n - 1} + f_{n - 2}$ with base cases $f_{0} = 0$ and $f_{1} = 1$ .

Define the polynomials ${F_{n}, n \geq 0}$ as $F_{n} (x) = \sum_{i = 0}^{n} f_{i} x^{i}$ .

For example, $F_{7} (x) = x + x^{2} + 2 x^{3} + 3 x^{4} + 5 x^{5} + 8 x^{6} + 13 x^{7}$ , and $F_{7} (11) = 268 357 683$ .

Let $n = 10^{15}$ . Find the sum $\sum_{x = 0}^{100} F_{n} (x)$ and give your answer modulo $1 307 674 368 000 (= 15!)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.