Find All Possible Stable Binary Arrays II

Hard

You receive 3 positive integers num_zeros, num_ones, and limit.

A binary array arr is called stable if:

The number of occurrences of 0 in arr is exactly num_zeros.
The number of occurrences of 1 in arr is exactly num_ones.
Each subarray of arr with a size greater than limit must contain at least one occurrence of both 0 and 1.

Produce an integer denoting the total number of stable binary arrays.

Since the answer may be very large, produce it modulo 10^9 + 7.

Example 1

Input: zero = 1, one = 1, limit = 2

Output: 2

Explanation: The two possible stable binary arrays are [1,0] and [0,1].

Example 2

Input: zero = 1, one = 2, limit = 1

Output: 1

Explanation: The only possible stable binary array is [1,0,1].

Example 3

Input: zero = 3, one = 3, limit = 2

Output: 14

Explanation: All the possible stable binary arrays are [0,0,1,0,1,1], [0,0,1,1,0,1], [0,1,0,0,1,1], [0,1,0,1,0,1], [0,1,0,1,1,0], [0,1,1,0,0,1], [0,1,1,0,1,0], [1,0,0,1,0,1], [1,0,0,1,1,0], [1,0,1,0,0,1], [1,0,1,0,1,0], [1,0,1,1,0,0], [1,1,0,0,1,0], and [1,1,0,1,0,0].

Find All Possible Stable Binary Arrays II

Constraints

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