Eccoci.

A window into a matrix is a contiguous sub matrix.

Consider a $16\times n$ matrix where every entry is either 0 or 1. Let $B(k,n)$ be the total number of these matrices such that the sum of the entries in every $2\times k$ window is $k$.

You are given that $B(2,4) = 65550$ and $B(3,9) \equiv 87273560 \pmod{1\,000\,000\,007}$.

Find $B(10^5,10^{16})$. Give your answer modulo $1\,000\,000\,007$.