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Problem 767

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Window into a Matrix II

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 $1\ 000\ 000\ 007$.

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矩阵窗口II

矩阵的窗口是指矩阵中一个连续的子矩阵。

考虑一个$16\times n$的矩阵，其中的元素均为$0$或$1$。
若矩阵中任意一个$2\times k$的窗口中元素之和都为$k$，记所有这样的矩阵数目为$B(k,n)$。

已知$B(2,4) = 65550$以及$B(3,9) \equiv 87273560 \pmod{1\ 000\ 000\ 007}$。

求$B(10^5,10^{16})$，并将你的答案对$1\ 000\ 000\ 007$取余。

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