Admissible Paths Through a Grid

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 $P(5)=25$, $P(16)=596994440$ and $P(1000) \bmod 1,000,000,007 = 341920854$.

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