Bounded Sequences

Problem 319

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

• x1 = 2
• for all 1 < in : xi-1 < xi
• for all i and j with 1 ≤ i, jn : (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.