Unique Strings
Description
Let's say that two strings $a$ and $b$ are equal if you can get the string $b$ by cyclically shifting string $a$. For example, the strings 0100110 and 1100100 are equal, while 1010 and 1100 are not.
You are given a binary string $s$ of length $n$. Its first $c$ characters are 1-s, and its last $n - c$ characters are 0-s.
In one operation, you can replace one 0 with 1.
Calculate the number of unique strings you can get using no more than $k$ operations. Since the answer may be too large, print it modulo $10^9 + 7$.
The first and only line contains three integers $n$, $c$ and $k$ ($1 \le n \le 3000$; $1 \le c \le n$; $0 \le k \le n - c$) — the length of string $s$, the length of prefix of 1-s and the maximum number of operations.
Print the single integer — the number of unique strings you can achieve performing no more than $k$ operations, modulo $10^9 + 7$.
Input
The first and only line contains three integers $n$, $c$ and $k$ ($1 \le n \le 3000$; $1 \le c \le n$; $0 \le k \le n - c$) — the length of string $s$, the length of prefix of 1-s and the maximum number of operations.
Output
Print the single integer — the number of unique strings you can achieve performing no more than $k$ operations, modulo $10^9 + 7$.
1 1 0
3 1 2
5 1 1
6 2 2
24 3 11
1
3
3
7
498062
Note
In the first test case, the only possible string is 1.
In the second test case, the possible strings are: 100, 110, and 111. String 101 is equal to 110, so we don't count it.
In the third test case, the possible strings are: 10000, 11000, 10100. String 10010 is equal to 10100, and 10001 is equal to 11000.