<p>You are given 3 positive integers <code>zero</code>, <code>one</code>, and <code>limit</code>.</p>

<p>A <span data-keyword="binary-array">binary array</span> <code>arr</code> is called <strong>stable</strong> if:</p>

<ul>
	<li>The number of occurrences of 0 in <code>arr</code> is <strong>exactly </strong><code>zero</code>.</li>
	<li>The number of occurrences of 1 in <code>arr</code> is <strong>exactly</strong> <code>one</code>.</li>
	<li>Each <span data-keyword="subarray-nonempty">subarray</span> of <code>arr</code> with a size greater than <code>limit</code> must contain <strong>both </strong>0 and 1.</li>
</ul>

<p>Return the <em>total</em> number of <strong>stable</strong> binary arrays.</p>

<p>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">zero = 1, one = 1, limit = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong></p>

<p>The two possible stable binary arrays are <code>[1,0]</code> and <code>[0,1]</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">zero = 1, one = 2, limit = 1</span></p>

<p><strong>Output:</strong> <span class="example-io">1</span></p>

<p><strong>Explanation:</strong></p>

<p>The only possible stable binary array is <code>[1,0,1]</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">zero = 3, one = 3, limit = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">14</span></p>

<p><strong>Explanation:</strong></p>

<p>All the possible stable binary arrays are <code>[0,0,1,0,1,1]</code>, <code>[0,0,1,1,0,1]</code>, <code>[0,1,0,0,1,1]</code>, <code>[0,1,0,1,0,1]</code>, <code>[0,1,0,1,1,0]</code>, <code>[0,1,1,0,0,1]</code>, <code>[0,1,1,0,1,0]</code>, <code>[1,0,0,1,0,1]</code>, <code>[1,0,0,1,1,0]</code>, <code>[1,0,1,0,0,1]</code>, <code>[1,0,1,0,1,0]</code>, <code>[1,0,1,1,0,0]</code>, <code>[1,1,0,0,1,0]</code>, and <code>[1,1,0,1,0,0]</code>.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= zero, one, limit &lt;= 1000</code></li>
</ul>

class Solution:
    def numberOfStableArrays(self, zero: int, one: int, limit: int) -> int:
        mod = 10**9+7
        @lru_cache(None)
        def part(n,k): # partition n into k numbers, 1<=xi<=limit
            if n<k or n>k*limit: return 0
            if k==n or k==1 or limit==1: return 1
            n -= k # each part >=0,((1-z^lim)/(1-z))^k
            # c(-k,n)-c(k,1)c(-k,n-lim)+..(-1)^ic(k,i)c(-k,n-i*lim) 
            t = comb(n+k-1,n)
            i = 1
            while n>=i*limit:
                nn = n-i*limit
                q = comb(k,i)*comb(nn+k-1,nn)%mod
                if i&1: t -= q
                else: t += q
                i += 1
            return t%mod
        #
        n1,n2 = min(zero,one),max(zero,one)
        total = 0
        for i in range((n1-1)//limit+1,n1+1):
            p = part(n1,i)
            total += (part(n2,i-1)+part(n2,i)*2+part(n2,i+1))*p
        return total%mod        
16900