A15505. Carry Bit

Let $f(x,y)$ be the number of carries of $x+y$ in binary (i. e. $f(x,y)=g(x)+g(y)-g(x+y)$ , where $g(x)$ is the number of ones in the binary representation of $x$ ).

Given two integers $n$ and $k$ , find the number of ordered pairs $(a,b)$ such that $0 \leq a,b < 2^n$ , and $f(a,b)$ equals $k$ . Note that for $a\ne b$ , $(a,b)$ and $(b,a)$ are considered as two different pairs.

As this number may be large, output it modulo $10^9+7$ .

The only line of each test contains two integers $n$ and $k$ ( $0\leq k<n\leq 10^6$ ).

Output a single integer — the answer modulo $10^9+7$ .

Here are some examples for understanding carries:

$$ \begin{aligned} &\begin{array}{r} 1_{\ \ }1_{\ \ }1\\ +\ _{1}1_{\ \ }0_{\ \ }0\\ \hline \ 1_{\ \ }0_{\ \ }1_{\ \ }1 \end{array} &\begin{array}{r} \ 1_{\ \ }0_{\ \ }1\\ +\ _{\ \ }0_{\ \ }0_{1}1\\ \hline \ 0_{\ \ }1_{\ \ }1_{\ \ }0 \end{array} & &\begin{array}{r} \ 1_{\ \ }0_{\ \ }1\\ +\ _{1}0_{1}1_{1}1\\ \hline \ 1_{\ \ }0_{\ \ }0_{\ \ }0 \end{array} \end{aligned} $$

So $f(7,4)=1$ , $f(5,1)=1$ and $f(5,3)=3$ .

In the first test case, all the pairs meeting the constraints are $(1,1),(1,5),(2,2),(2,3),(3,2),(4,4),(4,5),(4,6),(4,7),(5,1),(5,4),(5,6),(6,4),(6,5),(7,4)$.