MathDB
2017 Guts #28: Weird sequence and sum

Source:

February 21, 2017
algebra

Problem Statement

Let …,a−1,a0,a1,a2,…\dots, a_{-1}, a_0, a_1, a_2, \dots be a sequence of positive integers satisfying the folloring relations: an=0a_n = 0 for n<0n < 0, a0=1a_0 = 1, and for n≥1n \ge 1, an=an−1+2(n−1)an−2+9(n−1)(n−2)an−3+8(n−1)(n−2)(n−3)an−4.a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}. Compute ∑n≥010nann!.\sum_{n \ge 0} \frac{10^n a_n}{n!}.
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