AT_agc070_c [AGC070C] No Streak

Alice and Bob played rock-paper-scissors $ N $ times. Each round of rock-paper-scissors results in “Alice’s win,” “Bob’s win,” or “Draw.” Among all possible sequences of results of $ N $ rounds, how many satisfy the following conditions? Find the count modulo $ \bf{1000000007 = 10^9 + 7} $ . - Among the $ N $ rounds, Alice won exactly $ A $ times and Bob won exactly $ B $ times. - Alice never won twice in a row without intervening draws. - Bob never won twice in a row without intervening draws. - At no point did Bob’s number of wins exceed Alice’s number of wins. Formally, for all $ 1 \leq i \leq N $ , “the number of times Alice has won up through round $ i $ ” is at least “the number of times Bob has won up through round $ i $ .”

The input is given from Standard Input in the following format: > $ N $ $ A $ $ B $

Print the number, modulo $ (10^9 + 7) $ , of sequences of results of $ N $ rounds that satisfy the conditions.

### Sample Explanation 1 For example, the following sequence of $ N $ rounds satisfies the conditions: - Round $ 1 $ : Alice wins. - Round $ 2 $ : Bob wins. - Round $ 3 $ : Alice wins. - Round $ 4 $ : Draw. - Round $ 5 $ : Bob wins. On the other hand, the following sequence of $ N $ rounds does **not** satisfy the conditions, because after round $ 4 $ , Alice’s number of wins $ (=1) $ is less than Bob’s number of wins $ (=2) $ : - Round $ 1 $ : Alice wins. - Round $ 2 $ : Bob wins. - Round $ 3 $ : Draw. - Round $ 4 $ : Bob wins. - Round $ 5 $ : Alice wins. There are five sequences in total that satisfy the conditions. ### Sample Explanation 2 Remember to find the count modulo $ (10^9 + 7) $ . ### Constraints - $ 2 \leq N \leq 2 \times 10^7 $ - $ 1 \leq B \leq A $ - $ A + B \leq N $ - $ N $ , $ A $ , and $ B $ are integers.