Eccoci.

Let's call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares.
For example, $(9, 16)$ is inadmissible, while $(0, 4)$, $(3, 1)$ and $(9, 4)$ are not.

Consider a path from point $(x_1, y_1)$ to point $(x_2, y_2)$ using only unit steps north or east.
Let's call such a path admissible if none of its intermediate points are inadmissible.

Let $P(n)$ be the number of admissible paths from $(0, 0)$ to $(n, n)$.
It can be verified that $P(5) = 252$, $P(16) = 596994440$ and $P(1000) \bmod 1\,000\,000\,007 = 341920854$.

Find $P(10\,000\,000) \bmod 1\,000\,000\,007$.