<p>You have a pointer at index <code>0</code> in an array of size <code>arrLen</code>. At each step, you can move 1 position to the left, 1 position to the right in the array, or stay in the same place (The pointer should not be placed outside the array at any time).</p>

<p>Given two integers <code>steps</code> and <code>arrLen</code>, return the number of ways such that your pointer is still at index <code>0</code> after <strong>exactly</strong> <code>steps</code> steps. Since the answer may be too 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>

<pre>
<strong>Input:</strong> steps = 3, arrLen = 2
<strong>Output:</strong> 4
<strong>Explanation: </strong>There are 4 differents ways to stay at index 0 after 3 steps.
Right, Left, Stay
Stay, Right, Left
Right, Stay, Left
Stay, Stay, Stay
</pre>

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

<pre>
<strong>Input:</strong> steps = 2, arrLen = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are 2 differents ways to stay at index 0 after 2 steps
Right, Left
Stay, Stay
</pre>

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

<pre>
<strong>Input:</strong> steps = 4, arrLen = 2
<strong>Output:</strong> 8
</pre>

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

<ul>
	<li><code>1 &lt;= steps &lt;= 500</code></li>
	<li><code>1 &lt;= arrLen &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def numWays(self, steps: int, arrLen: int) -> int:
        modulo = 1000000007

        # Calculate the maximum possible position on the staircase that can be reached
        max_len = min(arrLen, 1 + steps // 2)

        # Initialize an array to store the number of ways to reach each position on the staircase
        ways = [0] * (max_len + 1)

        # Set the base case: There is one way to stay at the initial position
        ways[0] = 1

        # Iterate through the number of steps taken
        for i in range(steps):
            left = 0

            # Iterate through possible positions on the staircase, considering bounds
            for j in range(min(max_len, i + 2, steps - i + 3)):
                # Calculate the number of ways to reach the current position `j`
                # using the dynamic programming recurrence relation and considering the `left` value
                left, ways[j] = ways[j], (ways[j] + left + ways[j + 1]) % modulo

        # The `ways[0]` value represents the number of ways to reach the top of the staircase
        return ways[0]
        
17063