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

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

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 ${10}^9$.