CF1957E Carousel of Combinations

You are given an integer $ n $ . The function $ C(i,k) $ represents the number of distinct ways you can select $ k $ distinct numbers from the set { $ 1, 2, \ldots, i $ } and arrange them in a circle $ ^\dagger $ . Find the value of $$ \sum\limits_{i=1}^n \sum\limits_{j=1}^i \left( C(i,j) \bmod j \right) $$ Here, the operation $ x \\bmod y $ denotes the remainder from dividing $ x $ by $ y $ .

Since this value can be very large, find it modulo $ 10^9+7 $ .

$ ^\\dagger $ In a circular arrangement, sequences are considered identical if one can be rotated to match the other. For instance, $ \[1, 2, 3\] $ and $ \[2, 3, 1\]$$$ are equivalent in a circle.

N/A

N/A

In the first test case, $ C(1,1) \bmod 1 = 0 $ . In the second test case: - $ C(1,1)=1 $ (the arrangements are: $ [1] $ ); - $ C(2,1)=2 $ (the arrangements are: $ [1] $ , $ [2] $ ); - $ C(2,2)=1 $ (the arrangements are: $ [1, 2] $ ); - $ C(3,1)=3 $ (the arrangements are: $ [1] $ , $ [2] $ , $ [3] $ ); - $ C(3,2)=3 $ (the arrangements are: $ [1, 2] $ , $ [2, 3] $ , $ [3, 1] $ ); - $ C(3,3)=2 $ (the arrangements are: $ [1, 2, 3] $ , $ [1, 3, 2] $ ). In total, $ \left(C(1,1) \bmod 1\right) + \left(C(2,1) \bmod 1\right) + \left(C(2,2) \bmod 2\right) + \left(C(3,1) \bmod 1\right) + \left(C(3,2) \bmod 2\right) + \left(C(3,3) \bmod 3\right) = 4 $ .