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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 $B(3,9)\equiv 87273560(\mathrm{mod}1000000007)$.

Find $B({10}^5,{10}^{16})$. Give your answer modulo $1000000007$.

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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$B(3,9)\equiv 87273560(\mathrm{mod}1000000007)$。

求$B({10}^5,{10}^{16})$，并将你的答案对$1000000007$取余。

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