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Polynomials of Fibonacci numbers

Problem 435

Published on Saturday, 7th September 2013, 04:00 pm; Solved by 362

The Fibonacci numbers {fn, n ≥ 0} are defined recursively as fn = fn-1 + fn-2 with base cases f0 = 0 and f1 = 1.

Define the polynomials {Fn, n ≥ 0} as Fn(x) = ∑fixi for 0 ≤ i ≤ n.

For example, F7(x) = x + x2 + 2x3 + 3x4 + 5x5 + 8x6 + 13x7, and F7(11) = 268357683.

Let n = 1015. Find the sum [∑0≤x≤100 Fn(x)] mod 1307674368000 (= 15!).