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Polynomials of Fibonacci numbers

Published on Saturday, 7th September 2013, 04:00 pm; Solved by 885;
Difficulty rating: 30%

Problem 435

The Fibonacci numbers $\{f_n, n \ge 0\}$ 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 $\{F_n, n \ge 0\}$ as $F_n(x) = \displaystyle{\sum_{i=0}^n f_i x^i}$.

For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\,357\,683$.

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