Eccoci.

Let $x_1, x_2, \dots, x_n$ be a sequence of length $n$ such that:

• $x_1 = 2$
• for all $1 \lt i \le n$: $x_{i - 1} \lt x_i$
• for all $i$ and $j$ with $1 \le i, j \le n$: $(x_i)^j \lt (x_j + 1)^i$.

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 $t(10^{10})$ and give your answer modulo $10^9$.