Best-of Match Winner

Albert and Bob often play board games and decide the winner with a best-of series.

An X-win series is a match format where the first person to win X games wins the series, and at most $2X-1$ games are played. Board games between the two children never end in a draw. For example, when $X = 3$, the following results are possible. A means Albert won the game, and B means Bob won the game.

• AAA: Albert wins the series 3-0. Games 4 and 5 are not needed, so they are not played.
• BAAA, ABAA, AABA: Albert wins the series 3-1. Game 5 is not needed, so it is not played.
• BBAAA, BAABA, BABAA, ABBAA, ABABA, AABBA: Albert wins the series 3-2.
• Besides these 10 results, swapping A and B gives 10 more results where Bob wins the series.

After the matches, Bob says the following. The final winner of the $M$-win series is Bob, who won $M$ games first. However, if the series had been an $N$-win series, Albert would have won. In other words, Albert reached $N$ wins first, but Bob reached $M$ wins first and became the final winner. Of course, $N < M$.

For example, when $M = 2$ and $N = 1$, the only result is ABB, where Albert wins the first game and Bob wins the other two.

Another example is $M = 3$ and $N = 2$, which has three possible results:

• AABBB: In a 2-win series, Albert would have won at game 2. In a 3-win series, Bob is the final winner.
• ABABB: In a 2-win series, Albert would have won at game 3. In a 3-win series, Bob is the final winner.
• BAABB: In a 2-win series, Albert would have won at game 3. In a 3-win series, Bob is the final winner.

Given $N$ and $M$, count the results in which Bob reaches $M$ wins and becomes the final winner, while Albert reaches $N$ wins first. The count can be very large, so print it modulo $10^9+7$.

The first line contains the number of test cases $T$.

Each test case is given on one line with two integers $N$ and $M$, separated by a space.

For each test case, print the answer on its own line.

• $1 \le T \le 10$
• $1 \le N < M \le 10^5$