Problem 319: Bounded Sequences

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Let x1, x2,..., xn be a sequence of length n such that:

x1 = 2

for all 1 < i ≤ n : xi-1 < xi

for all i and j with 1 ≤ i, j ≤ n : (xi) j < (xj + 1)i

There are only five such sequences of length 2, namely: {2,4}, {2,5}, {2,6}, {2,7} and {2,8}. There are 293 such sequences of length 5; three examples are given below: {2,5,11,25,55}, {2,6,14,36,88}, {2,8,22,64,181}.

Let t(n) denote the number of such sequences of length n. You are given that t(10) = 86195 and t(20) = 5227991891.

Find t(1010) and give your answer modulo 109.