## Polynomials of Fibonacci numbers

### Problem 435

Published on Saturday, 7th September 2013, 04:00 pm; Solved by 447; Difficulty rating: 30%The **Fibonacci numbers** {f_{n}, n ≥ 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 ≥ 0} as F_{n}(x) = ∑f_{i}x^{i} for 0 ≤ i ≤ n.

For example, F_{7}(x) = x + x^{2} + 2x^{3} + 3x^{4} + 5x^{5} + 8x^{6} + 13x^{7}, and F_{7}(11) = 268357683.

Let n = 10^{15}. Find the sum [∑_{0≤x≤100} F_{n}(x)] mod 1307674368000 (= 15!).